1.9 Trigonometric Functions

Helpful Tools for Calculus, Chapter 1: Functions and Relations

Previous Section: 1.8 Logarithmic Functions

Forms

Trigonometry is such a wide topic that it could have earned an entire chapter. However, for the purposes of this book I will assume there is a foundational knowledge present to trigonometry and the focus will be put on reviewing the more fundamental part of trigonometric functions.
Trigonometry is built on two base functions: sin⁡(θ) and cos⁡(θ). Trigonometric functions take an input in the form of an angle, so the variable θ is often used. On a right triangle like the following:

sin⁡(θ) represents the ratio of the side opposite of the angle and the hypotenuse. That is to say, sin⁡(θ)= . Because of this it has a maximum value of 1 since the sides of a triangle cannot be larger than the hypotenuse.
Similarly, cos⁡(θ)= .

Example 1: On the given triangle, what is sin⁡(θ)?

Solution: sin⁡(θ)=opposite/hypotenuse= 8/10= 4/5

Example 2: Draw a triangle so that cos⁡(θ)= (-4)/7.
Solution: Since cos⁡(θ)= adjacent/hypotenuse, we know that the adjacent value is -4 and thus the angle is facing left.

Example 3: If sin⁡(θ)= 5/13, find cos⁡(θ).
Solution: Let’s begin by drawing a picture of the triangle whose opposite side is 5 and has a hypotenuse of 13.

We can then use the Pythagorean Theorem to find the missing side of the right triangle.
5^2+b^2=13^2
25+b^2=169
b^2=144
b=12
Thus, we can say that cos⁡(θ)= 12/13

Angles

There are two main ways to measure an angle. There are 360^o in a circle and 2π radians in a circle. The easiest way to convert is to use the formula where π=180^o. Angles can surpass a full circle, but we don’t normally represent them as such because they can be ‘simplified’. Every full rotation (2π or 360^o) takes you back to the same position. A third type of angle, that isn’t used often, is Tau (τ). Tau is the equivalent of 2π, so, it is nice to think that tau is one full rotation as opposed to pi which is half a rotation.

Degrees	Radians	Tau
0	0	0
30	π/6	τ/12
45	π/4	τ/8
60	π/3	τ/6
90	π/2	τ/4
180	π	τ/2
270	3π/2	3τ/4
360	2π	τ
540	3π	3τ/2
720	4π	2τ

Example 4: Convert 50 ^o to radians.
Solution: 50 ∙ (π radians)/180= 5π/18 radians

Example 5: Convert 3π/7 radians to degrees.
Solution:

Example 6: Express 17π/6 radians as an equivalent angle measured from 0 to 2π.
Solution: 17π/6 is larger than 2π so we can subtract 2π to get an equivalent angle that is one less rotation.

Other Forms

There are four other common trigonometric functions, all of which are built upon the two base functions we have already discussed:

In order, these are pronounced tangent, cotangent, secant, and cosecant.

Solving a Trigonometric Function

In the four quadrants of the coordinate plane, seen to the right, there are different values associated with sine and cosine.
Cosine generally refers to the x-coordinate while sine is the y-coordinate on the plane.
The phrase “All Students Take Calculus” can help you to remember which ones are positive in the quadrants.

The unit circle is helpful in solving with some common and important angle inputs:

When solving sin⁡(π/3) we should find the angle π/3 on the circle, which is on the top right. Since sin⁡(θ) refers to the y-coordinate, we then take the second number in the parenthesis and sin⁡(π/3)= √3/2. The same strategies can be used for any other input. Recall that if the angle is larger than 2π or 360°, then we can find an equivalent angle by subtracting 2π or 360° until the angle is between 0 and 2π or 360°.

Example 7: Evaluate sin(5π/3)
Solution: Since sinθ represents the y-coordinate on the unit circle, we just need to find the angle 5π/3 which can be found on the bottom right of the circle, and the solution is the second coordinate. So sin(5π/3)=

Example 8: Evaluate cos(5π/4)
Solution: Since cosθ represents the x-coordinate on the unit circle, we just need to find the angle 5π/4 which can be found on the bottom left of the circle, and the solution is the second coordinate. So

Example 9: Evaluate tan(11π/6)
Solution: For the more trigonometric functions that aren’t sine or cosine, it helps to find the reference angle, then relate it to the base functions afterwards. 11π/6 is similar to π/6, but in the fourth quadrant. In this quadrant tangent is negative, so

Example 10: Evaluate sec((-π)/3)
Solution: Using a similar strategy to the previous example, we can begin by saying that

The reference angle for 5π/3 is π/3 and it is in the fourth quadrant so cosine (and therefore secant) is positive. So,

Example 11: Evaluate cot(π)
Solution:

Thus, cot(π) is undefined.

Graphs

When graphing the base trigonometric functions, we see that the values are cyclical. They have different starting values (as in f(0) is different for f(x)=sinx and f(x)=cosx) but the general shape is identical. These are wave functions because of the appearance of their graphs:

Below is a few images of a three-dimensional interpretation of sin(x) and cos(x). Cosine describes the side-to-side motion of a circle while sine represents the up-down motion of a circle. These images might look weird, but you can find a neat animation online that helps put this in perspective!

The other four trigonometric functions will have an infinite number of vertical asymptotes each because of the presence of either sinθ or cosθ in the denominator:

Domain

For sinx and cosx there are no domain restrictions. Any angle can be inputted into these functions. It is standard, however, to think of the general inputs as being between 0 and 2π even though that is not the domain.

For each of the other trigonometric functions there are an infinite number of domain restrictions. It is often easiest to describe them in terms of which function is causing those restrictions. For example, tan⁡(x) exists so long as cos⁡(x)≠0 which occurs when x=π/2+kπ where k∈Z.

Range

For sinx and cosx the range is limited to values between -1 and 1. This is because sinx and cosx are ratios of sides to a hypotenuse, where a side can never be larger than the hypotenuse. So, we say that -1≤sinx≤1 must be true. This means y∈[-1,1] for both base functions.

For tanx and cotx, there is no value it cannot be, which is evident from the graphs. The range for these can be written as y∈(-∞,∞).

Finally, for secx and cscx, the range is limited to be the opposite of sinx and cosx because of how they are defined. So for both secx and cscx, y∈(-∞,-1]∪[1,∞).

Helpful Trigonometric Rules

Applications

TO BE COMPLETED

Review

Trigonometric Functions are built on the relationship between the sides of a triangle and an angle in the triangle.
sin(x) and cos(x) are the two base trigonometric functions.