Trigonometry – Math Teacher's Resource Blog

Orbit and Rotation of Planet Earth

As we gaze across a beautiful valley or stare in awe at a distant mountain, it is easy to forget that we are on a spinning platform that is traveling on an elliptical orbit around the sun at an average speed of 66,600 miles per hour. I find this seemly unending journey truly amazing. In this post, I would like to take a look at some of the facts that mankind has learned about this journey.

Before Nicholas Copernicus (1474 – 1543), many people thought that the Sun, planets, and stars rotated about the Earth, and each planet in turn rotates on its own private circular arc. This complicated Earth centered view of nature became so entrenched that it became an article of faith in the Catholic Church. In fact, the Catholic Inquisition threated Galileo (1564 – 1642) with torture on the rack unless he publicly retracted his belief in the Sun centered circular orbit Copernican world system. Galileo publicly retracted his belief in the Copernican world view and was spared torture on the rack, but spent the remaining years of his life under house arrest.

Johannes Kepler (1571 – 1630) discovered three laws of planetary motion which is a relatively simple description of planetary motion. (You may find it helpful to read my post Demonstrating Dynamics in a Mathematical Model.) Kepler’s first law stated that the orbit of a planet around the Sun is an ellipse where the Sun is located at one of the two foci of the ellipse. An ellipse is a very special curve where every point P on the ellipse, the distance from P to one focus point plus the distance from P to the other focus point, is a constant. The diagram below shows an ellipse with foci at F₁ and F₂, length of major axis = 10 units, length of minor axis = 6 units, and center point at (0, 0). For very point P on an ellipse, the sum of the distances from point P to the two focus points equals the length of the major axis. As indicated in the diagram below, an ellipse can be drawn by first anchoring the endpoints of a length of string on a piece of paper or cardboard. Use a pencil to make the string taunt, and then trace the curve by keeping the string taunt as you move the pencil along the elliptical curve.

To better understand planetary orbits, it’s necessary to understand what we mean by the eccentricity of an ellipse. If a = half the length of the major axis, and c = the distance from the center to a focus point, then the eccentricity e of the ellipse = c/a. Thus elliptical eccentricity e ranges from 0 to 1. If e = 0, the ellipse is a circle, and if e = 1, the ellipse degenerates to a line segment with foci at the endpoints of the major axis. (By definition, the eccentricity of a parabola equals 1, and the eccentricity of a hyperbola is greater than 1.) The two diagrams below show eccentricity values for five ellipses where the ellipse and foci have the same color. Note that eccentricity approaches 1 as the foci approach the endpoints of the major axis. The eccentricity of the Earth’s orbit = 0.0167086. This is the reason, I suspect, that Copernicus thought the Earth’s orbit was circular, not elliptical. Since half the length of the major axis of the Earth’s elliptical orbit equals 149.6 million km, it follows that the Sun is 0.0167086*149.6 million km = 2.4996 million km from the center of the Earth’s orbit.

The two diagrams below show an exaggerated oval shape of the Earth’s yearly orbit around the Sun; the purpose is to draw your attention to key time periods in a year. Orbital dates can vary slightly from year to year, and therefore the dates shown in the diagrams are approximate. The following points describe the key time periods in Earth’s orbit:

At the point of perihelion, the Earth is at its closest point of 147.1 million km from the Sun. In northern latitudes, the direction of the Earth’s polar axis is tilted away from the Sun, which results is less direct sunlight and cooler average temperatures.
At the point of aphelion, the Earth is at its farthest point of 152.1 million km from the Sun. In northern latitudes, the direction of the Earth’s polar axis is tilted towards the Sun, which results in more direct sunlight and warmer average temperatures.
The equinoxes and solstices divide a year into approximately four equal time periods or seasons. At the fall and spring equinoxes, the Earth’s polar axis is perpendicular to the plane of the Earth’s orbit which results in equal periods of daylight and darkness. At the summer and winter solstices, the Earth’s polar axis is tilted towards or away from the Sun which results the longest and shortest days of the year.
At the point of perihelion, the Earth reaches its fastest orbital speed of 109,080 km/hour.
At the point of aphelion, the Earth reaches its slowest orbital speed of 105,480 km/hour.
The average or mean orbital speed of the Earth equals 107,200 km/hour or 66,600 mph.
It takes the Earth 365.256 363 004 days to orbit the Sun. Because of the extra 0.256 363 004 days in a year, it’s necessary to add an extra day to our calendar every four years in February. To be more specific, leap years occur in years that are multiples of 4 or 400, but not multiples of 100. Hence the years 2000 and 2400 are leap years, but the years 1800, 1900, 2100, 2200 and 2300 are not leap years. All other years that are multiples of 4 such as 1868, 1936 and 2016 are leap years.
In the diagrams below, note that seasons in the northern and southern hemispheres occur at opposite times of the year.

Everyone knows that the Earth does a daily rotation about its polar axis. Here are a few facts about the Earth’s rotation.

The Earth rotates in about 24 hours with respect to the Sun and once every 23 hours, 56 minutes and 4 seconds with respect to the stars.
The Earth’s rate of rotation rate is slowing with time. Atomic clocks have demonstrated that a modern-day is about 1.7 milliseconds longer than a day in 1900. (I doubt that this fact will be reported in the national news any time soon.)
In the northern hemisphere, the Earth rotates east towards the Sun in the morning hours and away from the Sun in the west in the evening hours. This is the reason that the folks in New York see the Sun about 4 hours before the folks in California.
Technically speaking, there is no such thing as sunrise and sunset. The Sun only appears to rise and set in the sky because of the rotation of the Earth. Buckminster Fuller who was an American architect (geodesic domes) and systems theorist suggested that we should the terms sunsight and sunclipse because the terms sunrise and sunset do not accurately describe what we observe.
The Earth’s rate of rotation is not constant. The true solar day is about 10 seconds longer at the point of perihelion and 10 seconds shorter at the point of aphelion.
At the equator, the Earth’s linear speed of rotation is 465.1 m/s, 1,674.4 km/h or 1,040.4 mph. At higher latitudes, the linear rate of rotation is reduced by a factor of Cos(angle of latitude). Example: The Kennedy Space Center is located 28.59° North latitude and has a linear rotation rate of 1,674.4Cos(28.59°) = 1,470.23 km/h = 913.56 mph.

I will close this post about an epiphany I experienced many years ago. As I recall, it was about March of 1975 when my neighbor Chuck Beck invited me into his back yard to view Sun spot activity. Chuck had placed his expensive Celestron telescope with an attached power cord and lens filter on his picnic table. As I adjusted a knob on the Celestron in order to keep the Sun in view, I had the same physical sensation in my legs as if I was riding a merry-go-round. I thought to myself, “Johnson you really ARE on a moving and spinning platform in space!”

Linear Transformation Rule to Reflect over Oblique Line y = mx + b

This post discusses how the linear transformation rule for reflecting a figure over an oblique line y = mx + b can be used to create learning opportunities for high school students at different grade levels. Initially one might think that this topic is too advanced for most high school students. In my opinion, it’s possible to create a variety of activities based on linear transformations that will give mathematical nourishment to high school students at many different grade levels. An understanding of the concept of using a linear transformation to change a 2D graphic object to another 2D graphic object can definitely benefit college linear algebra students and computer graphic programmers. You may find it helpful to read my post, Geometric Transformations to Practice Basic Skills and Introduce Fundamental Concepts.

The content of this post is based on my free handout, Reflection over Any Oblique Line. This handout covers the following topics and items:

The derivation of the liner transformation rule (p, q) → (r, s) that reflects a figure over the oblique line y = mx + b where both r and s are functions of p, q, m, and b.
The derivation of the linear transformation rule (p, q) → (r, s) that reflects a figure over the oblique line y = mx + b where both r and s are functions of p, q, b, and θ = Tan^-1(m).
Given the specific equation of a line y = mx + b, show different ways of finding a linear transformation rule to reflect a preimage figure over the line y = mx + b.
Graphic images showing the reflection images of various polygons over different oblique lines.
A 13-step algorithm for the TI-84 graphing calculator to draw preimage and image polygons under a linear transformation.

The linear transformation rule (p, s) → (r, s) for reflecting a figure over the oblique line y = mx + b where r and s are functions of p, q, m, and b is given below. Finding the linear transformation rule given equation y = mx + b involves substituting the values for m and b in the formulas below, and then using basic operations with fractions to simplify each of the six coefficients of the linear transformation rule. What a sneaky way to get kids to practice operations with fractions.

As shown in the handout, Reflection over Any Oblique Line, the linear transformation rule for reflecting over the line y = -2x + 4 is (p, q) → (-3/5p – 4/5q + 16/5, -4/5p + 3/5q + 8/5). When reflecting over the line y = 3/5x – 4, the linear transformation rule is (p, q) → (8/17p + 15/17q + 60/17, 15/17p – 8/17q – 100/17). The graph below shows the reflection images of polygons over the lines y = -2x + 4 and y = 3/5x – 4.

The linear transformation rule (p, s) → (r, s) for reflecting a figure over the oblique line y = mx + b where r and s are functions of p, q, b, and θ = Tan^-1(m) is shown below. Finding the linear transformation rule given the equation of the line of reflection equation y = mx + b involves using a calculator to find angle θ = Tan^-1(m), and then calculating each of the six coefficients of the linear transformation rule.

The handout, Reflection over Any Oblique Line, shows the derivations of the linear transformation rules for lines of reflection y = √(3)x – 4 and y = -4/5x + 4.

Line y = √(3)x – 4: θ = Tan^-1(√(3)) = 60° and b = -4. The corresponding linear transformation rule is (p, q) → (r, s) = (-0.5p + 0.866q + 3.464, 0.866p + 0.5q – 2).
Line y = -4/5x + 4: θ = Tan^-1(-4/5) = -38.66° and b = 4. The corresponding linear transformation rule is (p, q) → (r, s) = (0.2195p – 0.9756q + 1.9512, -0.9765p – 0.2195q + 2.4390).

The graph below shows the reflection images of a polygon over the lines y = √(3)x – 4 and y = -4/5x + 4.

Suggestions for activities that teachers might consider:

Give students a sheet of graph paper with the line of reflection and preimage polygon drawn. Also give them the equation of the line of reflection and the linear transformation rule corresponding to the equation of the line of reflection. Have students use the linear transformation rule to calculate the vertices of the image polygon, and then draw the image polygon. A completed assignment should include lists of the x-y coordinates of the vertices of the preimage and the image polygon.
Give students a sheet of graph paper with the line of reflection and preimage polygon drawn. Have students find the equation of the line of reflection in slope-intercept format, and the linear transformation rule corresponding to the equation of the line of reflection. Depending on the level and interest of the students, allow students to calculate transformation rule coefficients in terms of parameters m and b or θ and b. Then have students use the linear transformation rule to calculate the vertices of the image polygon and draw the image polygon. A completed assignment should include lists of the x-y coordinates of the vertices of the preimage and the image polygon.
Have upper level students derive the linear transformation rule with parameters m and b. As illustrated in the handout, Reflection over Any Oblique Line, explain and discuss the general strategy for deriving the linear transformation rule with parameter m and b. Derivation of the equations for a linear transformation only requires an understanding of concepts already encountered in their math classes. This is a great activity for promoting mathematical reasoning and presenting ideas in an organized manner to explain mathematical relationships.
Have upper level trig students derive the linear transformation rule with parameters θ and b. Using the handout, Reflection over Any Oblique Line, as a guide, review and discuss the trig identities needed to convert the six coefficients of a linear transformation with parameters m and b to the coefficients of a linear transformation with parameters θ and b. Students now have a need to use trig identities.
If students have access to the appropriate technology, have them do a project in which they graph the reflection image of a polygon over an oblique line y = mx + b. The completed project should include the following items: 1) A graph showing the line of reflection, preimage polygon, and image polygon. 2) The equation of the line of reflection. 3) A describtion of the linear transformation rule corresponding to the equation of the line of reflection. 4) A list of the x-y coordinates of the vertices of the preimage polygon. 5) A list of the x-y coordinates of the vertices of the image polygon. Students who have access to a TI-84 graphing calculator can use the 13-step algorithm given in the handout Reflection over Any Oblique Line.
The handout, Reflection over Any Oblique Line, shows how linear transformation rules for reflections over lines can be expressed in terms of matrix multiplication. After showing students matrix multiplication based transformation rules, they better understand why matrix multiplication is done the way it is. Programmers use matrix multiplication to perform 2D and 3D transformations of objects on a computer screen. Computer video cards are optimized to perform millions of matrix multiplications per second.
Recruit a team of computer programming geeks in your school to write a program that calculates the x-y coordinates of a reflection image of a figure over any oblique line y = mx + b, and then graph the line of reflection, the preimage figure, and the image figure. The set of features the program could offer is limited only by the ability and imagination of the programmers.

Addition, Subtraction and Vectors

Early in a beginning algebra course, students are taught how to add and subtract signed numbers. Addition of signed numbers is taught by giving students a set of rules that they can follow to get the right answer. Some number line diagrams are thrown in to illustrate addition of signed numbers. Subtraction of signed numbers is usually taught by the “add the opposite” rule. This rule sounds plausible, however, it does not give students any insight into what subtraction is all about from a geometric point of view. You might ask, “Isn’t learning how to apply a set of rules to get the answer sufficient?” In my view, it’s not. As I stated several times in previous posts, whenever possible, students should understand a math concept from both an algebraic and geometric point of view. The primary purpose of this post to show how I have used vector diagrams to illustrate addition and subtraction of real numbers and 2D vectors. I will also discuss geometric interpretations of expressions and relationships such as (a + b)/2, |x – 4| < 5, and |x – 5| > 7.

Based on my own experience, most kids quickly learn how to apply the rules for adding and subtracting positive and negative numbers. When simplifying polynomial expressions, rational polynomial expressions, and polynomial long division, most mistakes occur at the step where a negative number is subtracted. Students are generally pretty good at adding signed numbers, but a little weak when it comes to subtracting signed numbers. This is why my favorite high school math teacher, Vivian Jones, said, “When subtracting a number, just change the sign of the number and add”. This is what I tell my college algebra students when they do polynomial long division. As taken from a developmental algebra textbook, the rules for adding and subtracting signed numbers are as follows:

Adding Two Real Numbers a and b

If a and b have the same sign, add their absolute values. Use the common sign of a and b as the sign of the answer.
If a and b have different signs, subtract their absolute values. The sign of the answer is the sign of the number that has the largest absolute value.

Subtracting Two Real Numbers a and b

a – b = a + (-b)
In other words, a – b equals a plus the opposite of b

Vector diagrams similar to diagrams A, B and C below are used to illustrate how the addition and subtraction rules for signed numbers work. The bottom half of diagram A shows that two negatives do not necessarily make a positive.

Diagrams B and C illustrate the subtraction rule for signed numbers. The problem with the subtraction rule and the corresponding vector diagram is that they don’t give us any insight into what subtraction is all about from a geometric point of view.

Diagrams D and E below illustrate the idea that subtracting two real numbers gives us the directed distance between two reals numbers. The concept of directed distance is fundamental in understanding subtraction from a geometric point of view. For real numbers a and b, the directed distance from b to a = a – b, and the directed distance from a to b = b – a. The positive distance or just the distance from a to b = |a – b| = |b – a|. The distance concept is related to many concepts in mathematics such as the amount of change in a variable, how much a data value deviates from some fixed constant, margin of error, etc. Notice that the subtraction problems in diagrams D and E are the same problems in diagrams B and C. After comparing diagrams B and C with diagrams D and E, it becomes clear that diagrams D and E give us a much better way to understand subtraction from a geometric point of view.

We will now take a look at 2D vectors which are essential in understanding a variety of concepts in math and physics. We can treat 2D vectors as line segments that have the properties of length and direction. Any two vectors are equivalent if and only if they have the same length and direction. In this post, 2D vectors will be denoted by bold face capital letters, and a pair of vertical absolute value bars will denote the length or magnitude of a vector. The arrow at one tip of a segment indicates the direction the vector. Geometric rays have infinite length, and therefore 2D vectors are not geometric rays. In passing, I will mention that 2D vectors are just a special case of an abstract mathematical object named vector. If you want to stretch your mind, take a course in infinite dimensional vector spaces. The student and teacher versions of my free handout, Introduction to Vectors, can be downloaded by going to the trigonometry section of our instructional content page.

The graph below shows how two 2D vectors are added. All vectors drawn in the same color are equal to each other because they have the same length and direction. All 2D vectors can be represented by a pair of real numbers of the form < x, y > where x and y equal the x-component and y-components of the vector. Knowing the x-y components of a vector, it’s easy, at least for trig students, to calculate the vector’s magnitude and direction. Note the following as you study the graph:

A 2D vector can be expressed as the sum of its x-component and y-component. Example: Vector A = < -7, 0 > + < 0, 3 > = < -7, 3>.
To find the x-y components of a vector, start at the tail of the vector and count the number of spaces left/right and the number of spaces up/down to the head of the vector.
The x-y components of any two equivalent vectors are equal.
The tail of the second vector in a vector sum is located at the tip of the first vector.
Vector addition is commutative. In other words, it makes no difference in what order vectors are added.
Any two equivalent vector pairs will always result in equivalent vector sums.
The theorem of Pythagoras is used to calculate the length or magnitude of a vector.
The inverse tangent function, Tan^-1(x), is used to calculate the direction of a vector.
Vectors A, B and S = A + B = B + A in the diagram have the following properties:
- A = < -7, 3 >, |A| = √(58) units, and direction of A ≈ 156.80⁰
- B = < 3, 5 >, |B| = √(34) units, and direction of B ≈ 59.04⁰
- S = < -4, 8 >, |S| = √(80) units, and direction of S ≈ 116.57⁰
- |A| + |B| > |S|

The graph below shows how two 2D vectors are subtracted. Vector subtraction gives us a vector that represents a difference vector between the tips of the two vectors. Vector A – B has its tail at the tip of B and its head at the tip of A. Vector B – A has its tail at the tip of A and its head at the tip of B. All vectors drawn in the same color are equal to each other because they have the same length and direction. Note the following as you study the graph:

The difference vector connects the tip of one vector to the tip of the other vector.
Vector subtraction is not commutative. Vectors A – B and B – A have the same length, but they point in opposite directions; they are vector opposites.
Vectors A, B, A – B, B – A have the following properties:
- A = < 6, 2 >, |A| = √(40) units, and the direction of A ≈ 156.80⁰
- B = < 2, 9 >, |B| = √(85) units, and the direction of B ≈ 59.04⁰
- A – B = < 4, -7 >, |A – B| = √(65), and the direction of A – B ≈ 299.75⁰
- B – A = < -4, 7 >, |B – A| = √(65), and the direction of B – A ≈ 119.75⁰

Once a student understands addition and subtraction from a geometric point of view, many math problems become much easier to solve. Consider the three routine math problems shown below.

Problem 1: Suppose the IQ score I of a person is in the normal range if the IQ score deviates from 100 by 10 points or less. What interval on a number line and inequality describes a normal IQ score?

Solution: The expression |I – 100| gives us the positive distance of the variable I from 100. Therefore the range of normal IQ scores is described by the inequality |I – 100| ≤10.

Problem 2: A part will fail inspection if its diameter d deviates from 2.5 cm by more than 0.001 cm. What interval on a number line and inequality describes the rejection region?

Solution: A part will fail inspection if the positive distance from 2.5 to d is more than 0.001 cm. Therefore the rejection region can be described by the inequality |d – 2.5| > 0.001. Of course, we tacitly assume that there are practical restrictions on values of d.

Problem 3: The graph of the closed interval [0.84, 2.68] is shown below. Find the following:

Length of the interval
Coordinate of the midpoint M
Radius of the interval
Write an inequality that describes the interval.

Solution: (Many of my elementary statistics students initially struggle with review problems like this.)

Length of interval = 2.68 – 0.84 = 1.84
Coordinate of midpoint M = (0.84 + 2.68)/2 = 1.76
Radius of the interval = 1.84/2 = 2.68 – 1.76 = 1.76 – 0.84 = 0.92
Inequality: |x – 1.76| ≤ 0.92

Miscellaneous facts I tell my students:

If you subtract a smaller number from a bigger number, the answer is positive.
If you subtract a bigger number from a smaller number, the answer is negative.
If you subtract a positive number from n, the answer is smaller than n.
If a positive influence is removed from your personal life, the quality of your personal life goes down.
If you subtract a negative number from n, the answer is bigger than n.
If you remove a negative influence from your person life, your personal life gets better.
If you add a negative number to n, the answer is smaller than n.
If a negative influence is introduced into your personal life, your personal life gets worse.
To find out how far apart two numbers are, subtract the numbers.
To find a number half way between two numbers, find the average of the numbers.

I will close this post with a true story about an epiphany I experienced early in my teaching career. The class was a regular high school geometry class. We were learning how to solve story problems involving complementary and supplementary angles. I could see that little Elmo (not his real name) was not getting the idea that if x is the measure of an acute angle, then 90 – x is the measure of the complement of the angle. So I asked Elmo a series of about 5 questions like: If an angle measures 20⁰, what is the degree measure of the complement of the angle? Elmo got every one of my questions right. I then asked Elmo the following question: If an acute angle measures n degrees, what is the degree measure of the complement of the angle? All I got from Elmo was a blank stare. I’m thinking to myself, why doesn’t he get it? Then it hit me. When asked the degree measure of the complement of a 65⁰, Elmo figured out how many degrees he needed to add to 65⁰ to get 90⁰. For me, this was an enlightening and humbling experience.

Division, Fractions, Proportional Parts, and Other Neat Stuff

After his freshman year in college, a former math student paid me a visit. Aaron (not his real name) indicated that his first year of college was a great experience. This was no surprise to me. I found Aaron to be a fairly perceptive individual who got along well with his high school classmates. Then Aaron said to me, “You know Mr. Johnson, after a year of college chemistry, I finally understand division.” He went on to explain how the concepts of division, fractions, decimals, ratios, rates, proportions and percent all fit together for him. In order to solve chemistry problems, Aaron was naturally motivated to learn and understand division and related concepts. Of course, I was surprised to hear Aaron make such a statement. He was an above average student who had attended a highly regarded local private school through junior high school. I then realized that if high school students like Aaron don’t really understand division, many of my students probably don’t really understand division either.

Based on numerous conversations with high school math teachers, in both public and private high schools, I have come to the conclusion that many high school students don’t really understand division and related concepts. It’s almost like a dirty little secret that high school math teachers don’t like to talk about in public. Many students have learned how to apply a set of rules to get the right answer, but they don’t really understand why a rule works or what the answer means. I see this all the time when I tutor developmental math students and above who are graduates of local area high schools.

Elementary, middle school, and high school teachers should NOT in any why interpret the above remarks to mean that they are not doing their job. In my view, teaching elementary and middle school students is far more difficult than teaching high school students. Of course, teaching high school math is not so easy either. If anyone thinks that they have a bullet proof method of teaching division and related concepts to a class of 30 typical middle school students, they should tell middle school teachers how to do it. I’m sure that middle school teachers would be ever grateful.

We constantly hear comments about how public education has deteriorated over the years. My now deceased father-in-law once complained to me that one of his office workers couldn’t figure out how many parts to ship to customers in proportion to the number of parts the customers ordered. (There were more parts ordered than parts in stock.). He said something like this, “Schools don’t teach kids the basics anymore. In my day, schools taught the fundamentals.” My father-in-law and many of his high school classmates learned basic math very well, but only 29% of the general population graduated from high school in the 1920’s. It’s my observation that today’s top students and athletes perform much better than the top students and athletes 40 or 50 years ago. As I see it, the perceived problem with modern education is due to the fact that public schools are attempting to reach a student population with a far greater range of abilities.

So what’s the purpose of this post? The purpose of this post is to share some of the methods that I have used in my classes to explain division and related concepts. Readers may find some of my examples and explanations to be ordinary, but other examples and explanations are different in that they are not found in textbooks. I invite readers to share some of their favorite methods that they use to explain division and related concepts to kids. I would love to do a post based on reader response. Readers should reply by sending an email to info@mathteachersresource.com. If your example has a graphic, please attach a file that contains the graphic. I will create the graphic for you if you give me a good description.

My first set of examples illustrates why the division rule for dividing two fractions gives us the correct answer. I assume that students already know how to apply the multiplication and division rules for fractions, and they know how to reduce a fraction. These examples reinforce the idea that dividing by a number is the same as multiplying by the reciprocal of the number.

Diagram A below illustrates the solution of following problem: If we have 12 pounds of hamburger and a meat loaf recipe that calls for 2 pounds of ground beef, how many meat loafs can we make? Solution: 12/2 = 12*(1/2) = 6 meat loafs.

Diagram B below illustrates the solution of following problem: If we have 12 pounds of hamburger, how many 1/4 pound hamburger patties can we make? Solution: 12/(1/4) =12*(4/1) = 12*4 = 48 hamburger patties.

Diagram C below illustrates the solution of following problem: If we have 12 pounds of hamburger, how many 3/4 pound hamburger patties can we make? Solution: 12/(3/4) = 12*(4/3) = 48/3= 16 hamburger patties. Alternatively, 12 pounds of hamburgers divided into 16 patties = 12/16 = 3/4 pounds per patty.

When I introduce right triangle trigonometry to my students, I give them the sides of a right triangle, and then we find the trig ratios of the two acute angles α and β of the triangle. Students quickly notice that Sin(α) = Cos(β). I want my students to not only understand the definitions of trig ratios and understand that trig function values are ratios, but I also want them to see that trig function values are percentages in disguise. The diagram below shows a right triangle, accurately drawn to scale, with 15⁰ and 75⁰ acute angles. Knowing only the acute angles of the right triangle and a scientific calculator, we can deduce the facts below. When students study the diagram, they often say, “When you look at it this way, it makes more sense.”

Cos(15⁰) = b/c ≈ 0.966. This tells us the length of the leg adjacent to a 15⁰ angle is always about 96.6% of the length of the hypotenuse.
Sin(15⁰) = a/c ≈ 0.259. This tells us the length of the leg opposite a 15⁰ angle is always about 25.9% of the length of the hypotenuse.
Tan(15⁰) = a/b ≈ 0.268. This tells us the length of the leg opposite a 15⁰ angle is always about 26.8% of the length of the leg adjacent to a 15⁰ angle.
Tan(75⁰) = b/a ≈ 3.73. This tells us the length of the leg opposite a 75⁰ angle is always about 373% of the length of the leg adjacent to a 75⁰ angle.

In the next two problems, an intuitive approach is used to solve proportional parts problems. This would have made my father-in-law happy.

Problem 1: Find the three angles of a triangle if the angles are in a ratio of 3 : 4 : 5.

Solution: Refer to the diagram below. The problem boils down to dividing 180⁰ into a ratio of 3 : 4 : 5 parts. Because 3 + 4 + 5 = 12, we divide 180 into 12 equal parts with each part equal to 180/12 = 15. The smallest angle = 3 * 15 = 45⁰. The next largest angle = 4* 15 = 60⁰ and largest angle = 5 * 15 = 75⁰. (I tell my students that I will only draw a diagram like the one below only one time because it takes too much time to draw the diagram. To my amazement, I once observed a student draw a diagram with well over 200 dots to solve a proportional parts homework problem. I’m not kidding.)

Problem 2: Customers A, B, C, D, and E have placed orders for 5, 7, 10, 20, and 40 widgets respectively. Since the company has only 60 widgets in stock, it was decided to immediately fill the orders in proportion to the number of widgets ordered, and ship the remaining widgets when they become available. How many widgets should be shipped to each customer?

Solution: Because 5 + 7 + 10 + 20 + 40 = 82, we divide 60 into 82 equal parts with each part equal to 60/82 = 0.732.

Customer A: Ship 5 x 0.732 = 3.66 or 4 widgets
Customer B: Ship 7 x 0.732 = 5.124 or 5 widgets
Customer C: Ship 10 * 0.732 = 7.32 or 7 widgets
Customer D: Ship 20 x 0.732 = 14.64 or 15 widgets
Customer E: Ship the remaining 29 widgets

The classic “working together” problems are difficult for beginning algebra students to understand. Before I show students the standard algebraic technique for solving a working together problem, I show them how to use ratios to solve a working together problem. The problem below is a typical working together problem.

Problem: It would take homeowner Bob 7 days to roof his garage and professional roofer Clyde could roof Bob’s garage in 3 days. If Bob and Clyde cooperate with each other, how many days should it take them to roof Bob’s garage if they work together? Assume that a work day equals 8 hours. The text box below shows the solution of the problem. I suppose Bob and Clyde would work over time the second day so that they could finish the job in 2 days. (There is always someone who will make an initial guess of 5 days!)

Now let’s take a look at a couple of percent problems from a slightly different point of view.

Problem 1: A sofa that normally sells for $2,799.95 is on sale at 20% off. Local sales tax rate equals 9%. Find the sale price of the sofa and the cost of the sofa with sales tax. The solutions are given in the text box below.

Problem 2: Brad and his lovely wife Angelina dined at an upscale restaurant to celebrate their 20^th wedding anniversary. The couple was fortunate to have a 15% discount coupon to help cover the $76.80 cost of their meal plus 9% sales tax. Since Brad was a high school math teacher, he thought that he had two options as to how he should apply the 15% discount. With option 1, he could take a 15% reduction of the cost of the meal plus sales tax. With option 2, he could first take a 15% discount on the cost of meal, and then pay sales tax on the discounted meal. With either option, Brad will leave a 20% tip. What option should Brad choose? (If Brad had stopped to really think about his options for a minute, he would have realized that it makes no difference what option he chooses.)

Option 1: Total cost = 0.85(1.09 x 76.80) = $71.16
Option 2: Total cost = 1.09(0.85 x 76.80) = $71.16
Total cost with tip = 1.20 x 71.16 = $85.39 or $85.00

I will close this post with some sample exercises that I have used to give students practice drawing valid conclusions and hopefully promote their mathematical reasoning ability. Each exercise involves a conditional relationship which is true or false depending on the values of the variables in the conditional statement. From a list of about 25 possible conclusions, student are to select all conclusions that are necessarily true. Given the long list of possible conclusions, most students don’t find these exercises so easy on first exposure. Each of the text boxes below shows a conditional statement and all valid conclusions which are contained in a list of possible conclusions.

When 0/0 or ∞/∞ Can Have a Value That Makes Sense

In a previous post, Why Division by Zero is Forbidden, I explained why a nonzero number divided by zero is undefined and why zero divided by zero gives us infinitely many answers.

Several comments from math teachers indicate that it is not easy to get this concept across to younger students so that they have a real understanding of the concept. I fully agree. From personal experience, I have found that students initially attain a rudimentary understanding of a math concept, but only time, practice working with the concept, and increased mental maturity can students gain a deeper understanding of a concept.

One reader, Ali-Carmen Houssney, wondered why an expression of the form 0 / 0 is called “indeterminate.” The purpose of this post is to discuss the indeterminate forms of the type 0 / 0 and ∞ / ∞. There are other indeterminate forms which you can look up online, but the forms 0 / 0 and ∞ / ∞ are the two major indeterminate forms that appear in calculus text books.

To get started, I need to discuss the concept of a limiting value of a function f(x) at some specific point x = k. The limit concept is a core concept in differential calculus. The limiting value of a function when x = k is simply a number or value that f(x) gets arbitrarily close to when x gets arbitrarily close to k or +∞. The definition of a limit of a function only requires that the input variable x gets arbitrarily close to some constant k and not necessarily equal to k. In most cases the limiting value of a function as x gets arbitrarily close to k is just f(k) itself. In other cases when f(k) results in the indeterminate form 0 / 0 or ∞ / ∞, the limiting value of f(x) when x gets arbitrarily close to k may exist or it may not exist. If the limiting value does exist, the limiting value will be a single finite real number or + ∞ which means that the function is increasing/decreasing without any upper/lower boundary. The text boxes and companion graphs below illustrate the limit concept for the functions y = f(x) = Sin(x) / x, y = g(x) = x/2^x and y = h(x) = Floor(x). All function outputs in the examples are rounded to 9 decimal places.

What’s important to notice in the examples below, is that values of x very close to k were selected to test whether or not the corresponding function values are very close to some fixed constant L, the limit of the function at x = k. A mathematically rigorous demonstration of the existence of the limit at x = k would show that there is always an open interval (a, b) about k in which all x in (a, b), except x = k, so that the corresponding function values are arbitrarily close to L, the limiting value of the function at x = k.

From a geometric point of view, it’s interesting to see why the limiting value of Sin(θ) / θ = 1 as θ approaches 0. Refer to graph D below. If central angle θ is in radians, the length of the arc on the unit circle that subtends angle θ equals θ units. Now imagine what happens as θ approaches 0; central angle θ approaches 0, Sin(θ) and length of the arc get closer and closer to each other. Hence, Sin(θ) / θ approaches 1 as θ approaches 0. This fundamental limit is used to derive the formulas for the derivative of all trigonometric functions. There is another interesting fact. If θ is converted to degrees, the limiting value of Sin(θ) /θ equals π/180 ≈ 0.017453293. You can easily convince yourself that this is true by setting the angle mode of your calculator to degrees and entering values of θ in the expression Sin(θ) /θ that are very close to 0.

In closing this post, I will mention L’Hôpital’s rule which is a very useful theorem for finding the limiting value of a function f(x) at x = k when f(k) results in the indeterminate form 0 / 0, ∞ / ∞, (-∞) / ∞, ∞ / (-∞) or (-∞) / (-∞). When f(k) results in indeterminate forms such as 0 * ∞, 1^∞, 0⁰, or ∞ – ∞, calculus students learn techniques for rewriting the expression for f(x) so that L’Hôpital’s rule can be applied. In order to use L’Hôpital’s rule, one needs to know how to find the derivative of a function.

Inverses of Functions and Relations

My previous post discussed the mathematical concepts of function and relation. Because the content of this post heavily depends on an understanding of the ideas presented in that post, you may find it helpful to read it before continuing.

The concept of the inverse of a relation is a natural extension of the important concept of a relation. The central idea is that an inverse relation is about reversing a relationship by exchanging variables, reversing/undoing an operation, or reversing/undoing a series of operations in a specific order. The following five questions and situations illustrate how a person uses the concept of the inverse of a relation to solve a problem.

(1) If we know a formula to convert Fahrenheit temperatures to Celsius, what formula converts Celsius temperatures to Fahrenheit?

(2) If we know a formula that tells us how to calculate the area of a circle from its radius, what formula will tell us how to calculate the radius of the circle from its area?

(3) A diner in a restaurant uses the restaurant’s menu function in inverse mode to determine what food items on the menu he/she can afford.

(4) A criminal investigator uses the one-to-one function that matches people with DNA molecules in inverse mode to match a sample of DNA molecules with a criminal.

(5) When solving for the sides and angles of a triangle, a trig student uses the inverse trig functions on his/her calculator to find the measure of an angle that has a specific trig function value.

The purpose of this post is to discuss inverse functions and relations when the matching rule is given by an x-y variable equation where both the domain and range is a subset of real numbers. These concepts will be discussed from algebraic and geometric points of view.

I will begin by looking at inverses of functions and relations from a geometric point of view. The two text boxes below summarize the geometric relationships between a relation and the inverse of a relation. The companion graphs illustrate the geometric relationships described in the text boxes. Notice that exchanging the variables in an equation gives us the equation of the inverse relation. These observations, of course, follow from the definition of the inverse of a relation, midpoint formula, definition of slope, and the fact that the product of the slopes of two perpendicular lines equals -1.

The text box below shows examples of elementary functions and the corresponding inverse relation which may or may not be a function. Notice that the inverse of the functions y = x² and y = |x| are relations, but not functions since y = x²and y = |x| are not one-to-one functions. As a reminder, the symbol √(x) means take the positive square root of x, and positive real numbers have a positive square root and a negative square root. Also note that the function y = Sin(x) is not one-to-one, and therefore the inverse relation is not a function. Calculators get around this problem by restricting the range of the function Sin^-1(x) to values that range from –π/2 to π/2.

The next part explains how I teach the inverse of trig functions y = Sin(x) and y = Cos(x). Initially, students struggle with the definitions of the inverse trig functions. Consider the equations listed in the edit box and graphs below. Because the trig functions are periodic, there are infinitely many solutions for each equation. Because the calculator keys Cos^-1(x) and Sin^-1(x) are function keys, the calculator should display only one of the infinitely possible output values. When x ranges from 0 to π, Cos(x) is one-to-one in adjacent quadrants I and II, and all possible output values of Cos(x) from -1 to 1 can be generated in quadrants I and II. Therefore Cos^-1(x) is a function if the output is restricted to range values from 0 to π radians. When x ranges from – π/2 to π/2, Sin(x) is one-to-one in adjacent quadrants I and IV, and all possible output values of Sin(x) from -1 to 1 can be generated in quadrants I and IV. Therefore Sin^-1(x) is a function if the output is restricted to range values from – π/2 to π/2 radians. I have my trig students find six solutions of simple trig equations. Example: Find six angles β in degrees in quadrant III, 3 positive and 3 negative, such that Cos(β) = -0.951056516. Round solutions to the nearest tenth of a degree.

I will conclude this post my showing you how I teach my students to find the inverse of a function when the function is composed of basic functions. The steps in the algorithm involve applying inverse operations in the reverse order of the order of operation rules. Exercises of this type reinforce concepts and are a good way to practice algebra skills. If you want to add some rigor to your course, have students check their solution by showing f(f^-1(x)) = f^-1(f(x)) = x. I remind students that an initial equation like x = y/(3y – 4) is an equation of the inverse relation, but it’s not expressed as a function of x. When a relationship is expressed as a function of x, we can graph the relation with a graphing utility. This is one of the reasons that we teach kids to solve an equation for a given variable. Sometimes I tell students to rearrange the equation for some variable because it makes more sense to them.

Useful tools from Math Teacher’s Resource:

• The graphs in my posts are created with my software, Basic Trig Functions. I think that you will find it very useful for teaching mathematical concepts in your classroom and developing custom instructional content. Relations can be entered as an explicitly defined function of x, an explicitly defined function of y, or as an implicitly defined x-y variable relation. Check it out at mathteachersresource.com/trigonometry.

• There are a wide variety of free handouts that teachers can use to create lessons or give to students as a handy reference handout. Among these handouts are Inverse Relations and Functions, Even and Odd Functions, and Relations and Functions Introduction handouts. Go to mathteachersresource.com/instructional-content to download MTR handouts. All content is available for immediate download. No sign-up required; no strings attached!

Comments Regarding My Previous Post:

• Some readers wanted to know the equation of the lead graph in my previous post. The equation of the graph is Cos(x) + Cos(y) >= 0.4 where both x and y range from -15 to 15. In view of the fact that Cos(x) is an even function, it should be no surprise that the graph has symmetry with respect to the x-axis, the y-axis, and the origin.

• The equation of the strange graph at the end of my previous post is 2xSin(3x) + 2y <= 3yCos(x + 2y) + 1. If you are skeptical, here are six solutions that you can plug into the equation to verify that the equation really does have solutions that satisfy the equality relationship. Just make sure that your calculator is in radian angle mode.

(-5.4, 5.195 577 636)

(5.5, 5.976 946 313)

(8.680 865 276, -5.2)

(-6.8, -6.786 215 284)

(0.578 827 17, -3)

(0.051 781 64, 5.8)

Comparing Circular Trig Functions with Hyperbolic Trig Functions

This post does a geometric compare and contrast of the circular functions Cos(x) and Sin(x) with the hyperbolic trig functions Cosh(x) and Sinh(x). I will do this by showing points of the form (Cos(t), Sin(t)) are points on the unit circle, and by showing points of the form (Cosh(t), Sinh(t)) are points on the unit hyperbola.

I will start the discussion by doing a quick review of radian angle measure and the definitions of the circular trigonometric functions Cos(θ), Sin(θ), and Tan(θ) where the input variable θ is the radian measure of a central angle of a circle with center at (0, 0) and radius equal to r. The text box below gives the definitions of the circular functions Cos(θ), Sin(θ), and Tan(θ) along with the radian measure of an angle and the formula for the area of a sector of a circle for angle θ. The formula for the area of a sector of a circle follows from the definition of radian angle measure and the fact that the area of a circular sector is proportional to the length of the arc that subtends the central angle.

The graph below illustrates the circular trig relationships in the text box above when r = 25 units, θ = α = 1.2870 radians and θ = β = -2.8578 radians. α, β, arc lengths and areas of circular sectors are rounded to 4 decimal places.

When the circular trig functions are defined in terms of the unit circle, r = 1, the input variable of the circular trig functions can be treated as an angle with angle measure = t radians, arc length = t length units, or time = t time units. This makes it possible to model periodic motion and periodic processes with circular trig functions. The text box below gives the definitions of Cos(t), Sin(t), and Tan(t) in terms of the unit circle. The unit circle graph illustrates the unit circle relationships when t = 2.58 units and t = -1.6 units. Notice that the area of the circular sector of arc length t = |t|/2 units².

It’s now time to take a look at the unit hyperbola and the hyperbolic trig functions x = Cosh(t) and y = Sinh(t). My previous post discussed the derivation, the graphs, and some identities for hyperbolic trig functions. If you have not read that post, I strongly suggest that you read it before continuing. The text box below summarizes the key points that are pertinent to this discussion. The graph illustrates the unit hyperbola relationships when t = 2.2 and t = -1.6. An understanding of integral calculus is required to understand the derivation of some hyperbolic relationships.

If you are interested in Einstein’s theory of special relatively, you will find The Geometry of Special Relatively by Tevian Dray a must read. The author shows an elegant way to express the Lorentz transformations in terms of Cosh(β) and Sinh(β) where β is implicitly defined by the equation Tanh(β) = v/c where v is some constant velocity and c is the speed of light. To my amazement, I learned that the Lorentz transformations are just hyperbolic rotations!

Dray’s book inspired me to think more deeply about the circular trig functions, hyperbolic trig functions, and the difference between the two geometries. Dray warns the reader that hyperbola geometry should not be confused with hyperbolic geometry which is the curved geometry of the two-dimensional unit hyperboloid. I will conclude this post with some of my personal observations about circular trig functions and hyperbolic trig functions, and finish with a wonderful example from Dray’s book that really clarified for me the difference between Euclidean geometry and hyperbola geometry.

The diagram below shows a right triangle in hyperbola geometry. The results are astonishing when you consider the definitions of Cos(θ), Sin(θ) and Tan(θ) for Euclidean right triangles. Of course, the triangle below would be impossible in Euclidean geometry, but possible in hyperbola geometry. Just think of the diagram as a clever way to picture time and space quantities in spacetime physics. In hyperbola geometry, a triangle is a right triangle if and only if the lengths of the sides satisfy the Pythagorean Theorem of hyperbola geometry, a²– b²= c² where a = c*Cosh(β), b = c*Sinh(β), and β is an angle of the triangle. From the formulas for the inverse hyperbolic trig functions, it follows that β = arcCosh(5/4) = arcSinh(3/4) = arcTanh(3/5) = 0.693147181. Notice that the hypotenuse of the hyperbolic right triangle is not the longest side.

Hyperbolic Functions Cosh(x), Sinh(x) and Tanh(x)

This post discusses how the function f(x) = e^x is used to create the hyperbolic trig functions Cosh(x), Sinh(x), and Tanh(x). Trig students immediately recognize the remarkable similarity between identities for the functions Cos(x), Sin(x), and Tan(x), and identities for the functions Cosh(x), Sinh(x), and Tanh(x). The hyperbolic trig functions have many important applications in many branches of mathematics and science. A couple of great examples are provided later in this post. I find these functions fun and interesting to play with, and I continue to find new ways of looking at and understanding these functions.

I will start the discussion by defining hyperbolic trig functions Cosh(x), Sinh(x), and Tanh(x) in terms of the functions y = f(x) = e^x / 2 and y = f(-x) = e^-x / 2 which are neither even nor odd. My last post discussed some of the properties and characteristics of even and odd functions. I concluded the post by showing how to create an even function and an odd function from any function which is not necessarily even or odd. If you have not read that post, you may want to read it before continuing. In view of the results obtained in my previous post, it follows that Cos(x) and Cosh(x) are even functions, and Sin(x), Sinh(x), Tan(x), and Tanh(x) are odd functions. The text box below gives the definitions of the three main hyperbolic trig functions. Graphs A, B, C, and D show the graphs of f(x), f(-x), Cos(x), Cosh(x), Sin(x), Sinh(x), and Tanh(x). As a reminder, the functions Cos(x), Sin(x), and Tan(x) are periodic, but the functions Cosh(x), Sinh(x), and Tanh(x) are not.

The text box below gives a comparison of some standard trigonometric identities and hyperbolic trig identities. Go online and check out the other hyperbolic trig identities. You will be amazed. The other day I found out that the derivative of an even function is an odd function, and the derivative of an odd function is an even function. Being able to understand and recognize even functions, odd functions, one-to-one functions, and the inverse of a function gives a student a whole new level of mathematical maturity and sophistication.

The textbox below shows the infinite Taylor series expansion of the functions Cos(x), Cosh(x), Sin(x), and Sinh(x). It’s interesting to see how close and yet very different the infinite series expansions of the functions are. Notice that the Taylor series expansion of Cos(x) and Cosh(x) are sums and differences of even functions! Also notice that the Taylor series expansion of Sin(x) and Sinh(x) are sums and differences of odd functions! The function e^xis the sum of even and odd functions, and therefore it’s neither even nor odd.

I find the infinite series expansion of the inverse functions for the circular trig functions and the hyperbolic trig functions very interesting. The similarities are striking. One can deduce whether or not the inverse of a function is an even or odd function by just doing a simple inspection the infinite series expansion of the function.

In doing research for this post, I discovered an interesting relationship between a catenary curve and a parabolic curve. Imagine a piece of chain, rope, or cable that is hanging from its endpoints to form a U-shaped curve. Galileo (1564 – 1642) thought that this U-shaped curve was parabolic. In 1691, the mathematicians Leibniz, Huygens and Johann Bernoulli showed that the U-shaped curve is described by the hyperbolic cosine function. Freely-hanging electric power cables, silk threads on a spider’s web, or suspension bridge cables have the U-shaped catenary curve. The Gateway Arch in St. Louis, Missouri is said to be an inverted catenary. All catenary curves are the result of sliding, stretching, rotating, or reflecting the graph of y = Cosh(x). Shown below are the equations and graphs of three curves. The results speak for themselves.

My next post will do a geometric compare and contrast of the circular functions Cos(x) and Sin(x) with the hyperbolic trig functions Cosh(x) and Sinh(x). I will do this by defining Cos(x) and Sin(x) in terms of the unit circle, and by defining Cosh(x) and Sinh(x) in terms of the unit hyperbola.

I will end this post my showing you the graph of the first eight terms of a Fourier approximation of a square sine wave. You will notice that the graph is the graph of an odd function and the Fourier approximation is the sum of eight odd functions. Those even and odd functions are everywhere!

Even and Odd Functions

The concepts of even and odd functions are usually introduced in advanced high school algebra, college algebra, trigonometry, or precalculus courses. Trig students learn how to apply the concepts of even and odd functions to simplify trigonometric expressions. Calculus students learn how to apply the concepts of even and odd functions to simplify the calculation of a definite integral. When students understand and can recognize even and odd functions, they are usually amazed how often these functions appear in application problems.

This post will discuss even and odd functions from both an algebraic and geometric point of view. Readers are encouraged to download my free handout Even and Odd Functions which is a handy reference that teachers can give to students.

I will start the discussion by describing even functions. The text box below gives a description of an even function from several points of view. Graph A illustrates what we mean when we say that a graph has symmetry with respect to the y-axis.

The text box below gives a description of an odd function from several points of view. Graph B illustrates what we mean when we say that a graph has symmetry with respect to the origin.

The text box below gives some basic observations about even and odd functions. My free handout Even and Odd Functions gives a more in depth list of the important properties of these functions.

The text box below shows the infinite Taylor series expansion of the function y = Cos(x). Graph C shows the graph of y = Cos(x) and the graph of the first five terms of the Taylor series expansion of Cos(x). Notice that the Taylor series expansion of Cos(x) is the sum and difference of even functions!

The text box below shows the infinite Taylor series expansion of the function y = Sin(x). Graph D shows the graph of y = Sin(x) and the graph of the first five terms of the Taylor series expansion of Sin(x). Notice that the Taylor series expansion of Sin(x) is the sum and difference of odd functions!

I will end this post by showing you how to create an even or odd function from any function y = f(x) that’s not necessarily even or odd. The text box below shows how and why this can be done. Graph E shows the results of creating an even function and an odd function from the function y = f(x) = 0.25(x – 4)² – 3Sin(x – 4) – 5. This result explains why the hyperbolic functions cosh(x) and sinh(x) are even and odd functions respectively.

More useful tools from Math Teacher’s Resource

• The graphs in this post were created with my software, Basic Trig Functions. I think that you will find it very useful for teaching mathematical concepts in your classroom and developing custom instructional content. Check it out at mathteachersresource.com/trigonometry.

• In addition to the Even and Odd Functions Handout linked in this post, Math Teacher’s Resource offers a wide variety of free math handouts, lessons, and exercises available at mathteachersresource.com/instructional-content. All content is available for immediate download. No sign-up required; no strings attached!

Explaining the Magical Property of Parabolic Reflectors – Part 2

My last post explains how parabolic reflectors have a magical property in that all incoming energy (light, sound, radio waves, etc.) traveling parallel to the axis of symmetry is reflected from the surface of the reflector to the focus point of the parabolic reflector. Conversely, all energy emanating from the focus point of a parabolic reflector is reflected to the surface of the reflector, and then in a direction that is parallel to its axis of symmetry. That post explained why parabolic reflectors work the way they do by verifying specific instances of the property.

This post will explain why parabolic reflectors have this property for any point on the reflecting surface of a parabolic reflector that is generated by rotating the graph of the equation y = kx²about its axis of symmetry. A principle of physics states that all energy in the form of particles or waves reflects off a flat surface in such a manner that the angle of incidence equals the angle of reflection. This post also demonstrates that the angle of incidence equals the angle of reflection where the flat surface is the tangent line at the point (x, y) on the graph of the equation y = kx².

All graphs in this post were created with my program Basic Trig Functions. Go to www.mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

I will begin my demonstration by presenting some observations about the relationship between the slope of a line and the tangent of an angle that has its terminal side contained in the line. Because of this relationship, the trigonometric identity tan(α – β) will be used to calculate the tangent of an angle from the slopes of the sides of the angle. Diagram (1) below illustrates these relationships.

The next part of my demonstration presents some observations about the relationship between the slope of a line and the tangent function value of an angle that has its terminal side contained in the line. Because one side of the angle is contained in a vertical line, the trigonometric identity cot(α – β) will be used to calculate the cotangent of an angle from the reciprocals of the slopes of the sides of the angle. Diagram (2) below illustrates these observations.

The last part of my demonstration shows that Tan(angle ABC) = Tan(angle FBE) for any point B = (x, kx²). This in turn justifies the claim that m(angle ABC) = m(angle FBE) since the Tan(θ) function is one-to-one for values of θ that range from 0 to π/2 radians. Diagram (3) below illustrates these relationships.

The key steps in the demonstration are given in the text boxes below. Most of the steps depend on the concepts illustrated in diagrams (1) and (2). The points, lines and angles in the demonstration refer to the points, lines, and angles in diagram (3) above.

It’s sufficient to show that the magical property is true for parabolas of the form y = kx² because translating, rotating, reflecting a graph over a line, and stretching the graph of a parabola preserves the property.