How to Find Exact Values for Trigonometric Functions

The unit circle is an excellent guide for memorizing common trigonometric values. However, there are often angles that are not typically memorized. We will thus need to use trigonometric identities in order to rewrite the expression in terms of angles that we know.

Preliminaries

In this article, we will be using the following trigonometric identities. Other identities can be found online or in textbooks.
Summation/difference
- $\cos(\theta \pm \phi )=\cos \theta \cos \phi \mp \sin \theta \sin \phi$
- $\sin(\theta \pm \phi )=\sin \theta \cos \phi \pm \cos \theta \sin \phi$
Half-angle
- $\cos {\frac {\theta }{2}}=\pm {\sqrt {\frac {1+\cos \theta }{2}}}$
- $\sin {\frac {\theta }{2}}=\pm {\sqrt {\frac {1-\cos \theta }{2}}}$

Steps

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1
Review the unit circle.^{[1]XResearch source} If you are not strong with the unit circle, it is important that you memorize the angles and understand for what quadrants are sine, cosine, and tangent positive and negative.

Part 1

Part 1 of 2:

Example 1

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1
Evaluate the following. The angle ${\frac {\pi }{12}}$ $How.com.vn English: {\frac {\pi }{12}}$ is not commonly found as an angle to memorize the sine and cosine of on the unit circle.
- $\cos {\frac {\pi }{12}}$
2
Write the expression in terms of common angles. We know the cosine and sine of common angles like ${\frac {\pi }{3}}$ $How.com.vn English: {\frac {\pi }{3}}$ and ${\frac {\pi }{4}}.$ $How.com.vn English: {\frac {\pi }{4}}.$ It will therefore be easier to deal with such angles.^{[2]XResearch source}
- $\cos {\frac {\pi }{12}}=\cos \left({\frac {\pi }{3}}-{\frac {\pi }{4}}\right)$
3
Use the sum/difference identity to separate the angles.^{[3]XResearch source}
- $\cos \left({\frac {\pi }{3}}-{\frac {\pi }{4}}\right)=\cos {\frac {\pi }{3}}\cos {\frac {\pi }{4}}+\sin {\frac {\pi }{3}}\sin {\frac {\pi }{4}}$
4
Evaluate and simplify.
- ${\frac {1}{2}}\cdot {\frac {\sqrt {2}}{2}}+{\frac {\sqrt {3}}{2}}\cdot {\frac {\sqrt {2}}{2}}={\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}$

Part 2

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Example 2

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1
Evaluate the following.
- $\sin {\frac {\pi }{8}}$
2
Write the expression in terms of common angles. Here, we recognize that ${\frac {\pi }{8}}$ $How.com.vn English: {\frac {\pi }{8}}$ is half of ${\frac {\pi }{4}}.$ $How.com.vn English: {\frac {\pi }{4}}.$ ^{[4]XResearch source}
- $\sin {\frac {\pi }{8}}=\sin \left({\frac {1}{2}}\cdot {\frac {\pi }{4}}\right)$
3
Use the half-angle identity.^{[5]XResearch source}
- $\sin \left({\frac {1}{2}}\cdot {\frac {\pi }{4}}\right)=\pm {\sqrt {\frac {1-\cos {\frac {\pi }{4}}}{2}}}$
4
Evaluate and simplify. The plus-minus on the square root allows for ambiguity in terms of which quadrant the angle is in. Since ${\frac {\pi }{8}}$ $How.com.vn English: {\frac {\pi }{8}}$ is in the first quadrant, the sine of that angle must be positive.
- ${\sqrt {\frac {1-\cos {\frac {\pi }{4}}}{2}}}={\frac {\sqrt {2-{\sqrt {2}}}}{2}}$

Community Q&A

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Question
How do I find the exact value of sine 600?
Donagan
Top Answerer
600° = 60° when considering trig functions. [600 - (3)(180) = 60] Sine 600° = sine 60° = 0.866.
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What does ASTC stand for in trigonometry?
Donagan
Top Answerer
It stands for the "all sine tangent cosine" rule. It is intended to remind us that all trig ratios are positive in the first quadrant of a graph; only the sine and cosecant are positive in the second quadrant; only the tangent and cotangent are positive in the third quadrant; and only the cosine and secant are positive in the fourth quadrant.
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What's the exact value of cosecant 135?
Donagan
Top Answerer
You can find exact trig functions by typing in (for example) "cosecant 135 degrees" into any search engine.
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