How To Find Exact Value Of Trig Functions Using Unit Circle

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The unit circumvolve is an fantabulous guide for memorizing mutual trigonometric values. However, at that place are oft angles that are non typically memorized. We will thus demand to employ trigonometric identities in lodge to rewrite the expression in terms of angles that we know.

Preliminaries

In this article, we volition be using the following trigonometric identities. Other identities can exist institute online or in textbooks.
Summation/difference
One-half-angle

1
Evaluate the following. The bending ${\frac {\pi }{12}}$ is non commonly found equally an bending to memorize the sine and cosine of on the unit of measurement circle.
- $\cos {\frac {\pi }{12}}$
2
3
Use the sum/difference identity to carve up the angles. ^[3]
- $\cos \left({\frac {\pi }{3}}-{\frac {\pi }{4}}\right)=\cos {\frac {\pi }{iii}}\cos {\frac {\pi }{iv}}+\sin {\frac {\pi }{3}}\sin {\frac {\pi }{4}}$
4
Evaluate and simplify.
- ${\frac {1}{2}}\cdot {\frac {\sqrt {2}}{two}}+{\frac {\sqrt {3}}{2}}\cdot {\frac {\sqrt {2}}{2}}={\frac {{\sqrt {2}}+{\sqrt {6}}}{iv}}$
Advertizing

one
Evaluate the post-obit.
- $\sin {\frac {\pi }{8}}$
ii
3
Use the half-bending identity. ^[5]
- $\sin \left({\frac {1}{2}}\cdot {\frac {\pi }{4}}\correct)=\pm {\sqrt {\frac {1-\cos {\frac {\pi }{4}}}{2}}}$
4
Evaluate and simplify. The plus-minus on the square root allows for ambivalence in terms of which quadrant the bending is in. Since ${\frac {\pi }{8}}$ is in the kickoff quadrant, the sine of that angle must exist positive.
- ${\sqrt {\frac {one-\cos {\frac {\pi }{four}}}{2}}}={\frac {\sqrt {2-{\sqrt {two}}}}{2}}$
Advertizement

Add New Question

Question

How do I notice the exact value of sine 600?

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

What does ASTC stand for in trigonometry?

Information technology stands for the "all sine tangent cosine" rule. Information technology is intended to remind united states that all trig ratios are positive in the first quadrant of a graph; only the sine and cosecant are positive in the second quadrant; but the tangent and cotangent are positive in the third quadrant; and only the cosine and secant are positive in the fourth quadrant.
Question

What's the exact value of cosecant 135?

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