How To Find The Exact Value Of Trig Functions

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

Preliminaries

In this article, we volition exist using the following trigonometric identities. Other identities can be found online or in textbooks.
Summation/divergence
Half-angle

1
Evaluate the following. The angle ${\frac {\pi }{12}}$ is non unremarkably found as an bending to memorize the sine and cosine of on the unit circle.
- $\cos {\frac {\pi }{12}}$
two
three
Employ the sum/divergence identity to separate the angles. ^[3]
- $\cos \left({\frac {\pi }{3}}-{\frac {\pi }{4}}\right)=\cos {\frac {\pi }{3}}\cos {\frac {\pi }{four}}+\sin {\frac {\pi }{3}}\sin {\frac {\pi }{4}}$
4
Evaluate and simplify.
- ${\frac {ane}{ii}}\cdot {\frac {\sqrt {2}}{2}}+{\frac {\sqrt {3}}{2}}\cdot {\frac {\sqrt {ii}}{two}}={\frac {{\sqrt {2}}+{\sqrt {six}}}{4}}$

1
Evaluate the following.
- $\sin {\frac {\pi }{eight}}$
2
iii
Use the half-angle identity. ^[v]
- $\sin \left({\frac {1}{2}}\cdot {\frac {\pi }{4}}\correct)=\pm {\sqrt {\frac {1-\cos {\frac {\pi }{iv}}}{2}}}$
4
Evaluate and simplify. The plus-minus on the foursquare 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.
- ${\sqrt {\frac {one-\cos {\frac {\pi }{4}}}{2}}}={\frac {\sqrt {2-{\sqrt {2}}}}{two}}$

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Question

How do I find the verbal value of sine 600?

600° = threescore° when because trig functions. [600 - (three)(180) = sixty] Sine 600° = sine 60° = 0.866.
Question

What does ASTC stand for in trigonometry?

It stands for the "all sine tangent cosine" dominion. Information technology is intended to remind the states that all trig ratios are positive in the first quadrant of a graph; just the sine and cosecant are positive in the second quadrant; just 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'due south the verbal value of cosecant 135?

You can find exact trig functions past typing in (for example) "cosecant 135 degrees" into any search engine.