## Teaching the Circular Sine and Cosine Functions

Students begin the study of trigonometry by learning how to solve for the sides and angles of a right triangle. Given any two sides of a right triangle, it is possible to solve for the length of the third side and the angle measure of the two acute angles. Likewise, given the measure of an acute angle and the length of one side, it is possible to find the lengths of the other two sides and the angle measure of the other acute angle. What makes this possible are the trigonometric sine, cosine and tangent functions. Before the availability of electronic calculators, the cosine, secant and cotangent functions were used to simplify certain paper and pencil calculations because it is much easier to multiply decimal numbers than divide decimal numbers.

After mastering right triangle trigonometry, students are taught radian angle measure and the six basic trig functions in terms of a circle with center at (0, 0) and radius r > 0. It is now possible to consider trigonometric function values with input values in degrees or radians such as Sin(56°), Csc(951°), Sin(-617π/3), Cos(225°), Sec(113π), Tan(155.296°) and Cot(-7.2). In the diagram below, α is a positive quadrant II angle and β is a negative quadrant III angle. The Cos, Sin, and Tan function values of α and β are shown below. For angle α, a circle with radius = 25 units and a point at (x1, y1) on the circle is used to find the function values. For angle β, a circle with radius = 41 units and a point at (x2, y2) on the circle is used to find the function values.

In this post, I will show teachers how they can use my program, Basic Trig Functions, to dynamically present the properties of circular trig functions. Similar demonstrations in my own classes give me a more effective way to teach the properties of these functions. Static graphs fail to capture the underlying dynamic properties of the functions. Basic Trig Functions is designed to be a tool to help teach a variety of core concepts in mathematics.

The objective of the first animation is to compare and contrast the properties of the functions y = Sin(x) and y = Cos(x) where x = the radian measure of an angle that ranges from 0 to 2π radians. The functions values are the changing x-y coordinates of a point on the unit circle as angle θ ranges from 0 to 2π radians. Students easily see when the function values are positive and negative as the point (x, y) moves from one quadrant to the next on the unit circle. After a little coaching, students can find function domain values where the value of a function equals -1, 0, or 1. What is shown in the animation is exactly what can be generated in a classroom with the program Basic Trig Functions. Of course, users have a variety of choices for setting output parameters. Mouse button clicks or button presses control program output rate.

The purpose of the second animation is to compare and constrast the properties of the functions y = 2Sin(x), y = Sin(x), and y = Sin(2x) where x = the radian measure of an angle that ranges from 0 to 2π radians. The animation naturally leads to a discussion of the concepts of amplitude and period of a function. For beginning students, it is not obvious that there is a difference in how function values for y = 2Sin(x) and y = Sin(2x) are calculated.

The third animation compares and constrasts the properties of the circular functions Cos(θ), Sin(θ), and Tan(θ) where the angle mode for θ can be set to exact radian, decimal radian, or degree measure. Angle input values are controlled by clicking convenient buttons. Users have options for setting the radius of the circle, setting the angle increment/decrement value, setting angle mode, and whether or not to draw a circular arc on the circle. The circular arc feature makes it much easier to explain radian angle measure because the length of the arc is displayed as angle θ changes.

Readers can download the free handouts Trig Summary, Unit Circle, Trig Exercises 1, and Trig Exercises 2 from our instructional content page. Trig Exercises 1 and Trig Exercises 2 give suggested student exercises which are designed to reinforce and integrate a number of key concepts. On first exposure to exercises of this type, teachers and students should do some of the exercises together.

Teaching Points:

• Instructors can have a student operate the program Basic Trig Functions during a class presentation. This allows instructors to stand near the projection screen, explain various concepts, and ask questions as the lesson progresses.

• Initially students will be confused because they are attempting to understand the generalized definitions of the trig functions in terms of right triangle trigonometry. Remind students that the generalized trig function definitions apply to both positive and negative angles of any size, not just the acute angles of a right triangle. Of course, the concept of a reference angle is useful in explaining the connection between right triangle trigonometry and the generalized definitions of the trig functions.

• To engage students, continually ask questions as the lesson progresses. It is more effective if a question is directed to a specific student.

• Conduct an experiment during the lesson by having students use a calculator to find Sin(20°), Sin(160°), Sin(200°), and Sin(340°). Some questions: What did you notice? Why are some sine function output values positive and other values negative? Use the inverse sine function to find six angles θ in quadrant II, 3 positive and 3 negative, such that Sin(θ) = 0.6156614753. Why does your calculator give you an error message when you enter Sin-1(1.25)? Then repeat the experiment with positive or negative angles and other trig functions.

The graphics in this blog were created with the program, Basic Trig Functions, which is offered by Math Teacher’s Resource. To view multiple screen shots of the program’s modules, go to www.mathteachersresource.com. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.