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.