how to find the exact value of a trigonometric function

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The unit circumvolve is an excellent guide for memorizing common trigonometric values. Still, there are often angles that are non typically memorized. We will thus need to use trigonometric identities in gild to rewrite the expression in terms of angles that we know.

Preliminaries

In this article, nosotros will be using the following trigonometric identities. Other identities can be plant online or in textbooks.
Summation/divergence
One-half-angle

i
Evaluate the following. The angle ${\frac {\pi }{12}}$ is not usually found as an bending to memorize the sine and cosine of on the unit of measurement circle.
- $\cos {\frac {\pi }{12}}$
2
3
Apply the sum/difference identity to dissever the angles. ^[iii]
- $\cos \left({\frac {\pi }{3}}-{\frac {\pi }{iv}}\correct)=\cos {\frac {\pi }{iii}}\cos {\frac {\pi }{4}}+\sin {\frac {\pi }{three}}\sin {\frac {\pi }{4}}$
four
Evaluate and simplify.
- ${\frac {i}{2}}\cdot {\frac {\sqrt {2}}{2}}+{\frac {\sqrt {3}}{2}}\cdot {\frac {\sqrt {two}}{two}}={\frac {{\sqrt {2}}+{\sqrt {6}}}{iv}}$

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one
Evaluate the following.
- $\sin {\frac {\pi }{viii}}$
ii
3
Use the half-angle identity. ^[5]
- $\sin \left({\frac {ane}{ii}}\cdot {\frac {\pi }{4}}\right)=\pm {\sqrt {\frac {one-\cos {\frac {\pi }{4}}}{ii}}}$
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 get-go quadrant, the sine of that angle must be positive.
- ${\sqrt {\frac {1-\cos {\frac {\pi }{four}}}{2}}}={\frac {\sqrt {two-{\sqrt {2}}}}{2}}$

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Question

How do I discover the exact value of sine 600?

600° = 60° when considering trig functions. [600 - (3)(180) = 60] Sine 600° = sine threescore° = 0.866.
Question

What does ASTC stand for in trigonometry?

Information technology stands for the "all sine tangent cosine" dominion. It is intended to remind united states of america that all trig ratios are positive in the get-go quadrant of a graph; only the sine and cosecant are positive in the second quadrant; only the tangent and cotangent are positive in the 3rd 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?

You lot can discover verbal trig functions past typing in (for case) "cosecant 135 degrees" into whatever search engine.