How to Find Exact Values for Trigonometric Functions

Co-authored by David Jia

Last Updated: October 8, 2024

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
- $\cos {\frac {\theta }{2}}=\pm {\sqrt {{\frac {1+\cos \theta }{2}}}}$
- $\sin {\frac {\theta }{2}}=\pm {\sqrt {{\frac {1-\cos \theta }{2}}}}$

Part 1

Part 1 of 3:

Learning the Unit Circle

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/f\/f9\/768px-Unit_circle_angles_color.svg.png\/460px-768px-Unit_circle_angles_color.svg.png","bigUrl":"\/images\/thumb\/f\/f9\/768px-Unit_circle_angles_color.svg.png\/560px-768px-Unit_circle_angles_color.svg.png","smallWidth":460,"smallHeight":460,"bigWidth":560,"bigHeight":560,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"http:\/\/en.wikipedia.org\/wiki\/Public_domain\">Public Domain<\/a><br>Details: From <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Unit_circle_angles_color.svg\">Wikimedia Commons<\/a><br>\n<\/p><\/div>"}

1

Review the unit circle. ^{[1]
X
Research 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 2

Part 2 of 3:

Example 1

Download Article

1
Evaluate the following. The angle ${\frac {\pi }{12}}$ 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 ${\frac {\pi }{3}}$ and ${\frac {\pi }{4}}.$ It will therefore be easier to deal with such angles.
- $\cos {\frac {\pi }{12}}=\cos \left({\frac {\pi }{3}}-{\frac {\pi }{4}}\right)$
3
Use the sum/difference identity to separate the angles.
- $\cos \left({\frac {\pi }{3}}-{\frac {\pi }{4}}\right)=\cos {\frac {\pi }{3}}\cos {\frac {\pi }{4}}+\sin {\frac {\pi }{3}}\sin {\frac {\pi }{4}}$
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 3

Part 3 of 3:

Example 2

Download Article

1
Evaluate the following.
- $\sin {\frac {\pi }{8}}$
2
Write the expression in terms of common angles. Here, we recognize that ${\frac {\pi }{8}}$ is half of ${\frac {\pi }{4}}.$ ^{[2]
X
Research source}
- $\sin {\frac {\pi }{8}}=\sin \left({\frac {1}{2}}\cdot {\frac {\pi }{4}}\right)$
3
Use the half-angle identity. ^{[3]
X
Research 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 ${\frac {\pi }{8}}$ 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

Search

Add New Question

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.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 62 Helpful 12
Question

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.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 8 Helpful 16
Question

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.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 83 Helpful 9

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

Tips

Submit a Tip

All tip submissions are carefully reviewed before being published

Name

Please provide your name and last initial

Submit

Thanks for submitting a tip for review!

[1]

[2]

[3]

Log in

How to Find Exact Values for Trigonometric Functions

Preliminaries

Steps

Learning the Unit Circle

Example 1

Example 2

Community Q&A

Tips

You Might Also Like

References

About This Article

Reader Success Stories

Did this article help you?

Quizzes

You Might Also Like

Featured Articles

Trending Articles

Featured Articles

Featured Articles

Watch Articles

Trending Articles

Quizzes

Explore Categories