Trig Identites

Trigonometry Tutorial

Trig Identites

Intro

This problem uses trigonometric identities to solve. It was found in the Princeton Review’s Math Level 2 SAT Subject Test book.

Sample Problem

If sin(θ) = 1/3 and (-π/4) < θ < (π/4), then cos(2θ) = ?

2/3


-7/9


-2/3


1


7/9


Solution

To approach this problem, look at the information given and what is needed to solve the problem.

You are given that for angle θ, the ratio of the side opposite to it to the hypotenuse of the triangle. This is equal to 1/3. You need to solve for angle θ to find the ratio of the adjacent angle to the hypotenuse (cosine) in a triangle whose angle is 2 times θ.

First, set solve for θ.
sinθ = 1/3
In order to isolate θ, you need to do the inverse of sin (may be written as sin^-1 as well).
(sinθ = 1/3)θ
The sine function cancels, so you are left with
θ = arcsin(1/3)
If you plug this into your calculator (remember to use degrees!), you get
θ = 19.47
Now you need to substitute θ for 19.47
cos(2θ) =
cos(2*19.47) =
This can be plugged into the calculator.
= 0.7777778
Since the answers are in fraction form, this is approximately equal to 7/9.



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