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Trigonometric Identities and Equations, Page 21

ExamplesPaper with a grade of A

Now that you have proved the Double-Angle and Half-Angle Identities, let’s go through some examples.

Example 1:

If sinθ = 1/3 and theta is in Quadrant I, find the following values:

a. sin 2theta

First, you will need to find costheta.

(sinθ)^2 + (cosθ)^2 = 1; (1/3)^2 + (cosθ)^2 = 1; 1/9 + (cosθ)^2 = 1; (cosθ)^2 = 1 - 1/9; (cosθ)^2 = 8/9; cosθ = sqrt(8/9) = 2sqrt(2)/3

Notice that since angle theta is in Quadrant I, you should only be interested in the positive value of costheta.

sin2θ = 2sinθ cosθ; sin2θ = 2(1/3)(2sqrt(2)/3) = 4sqrt(2)/9

b. cos 2theta

cos2θ = 1 - 2(sinθ)^2; cos2θ = 1 - 2(1/3)^2; cos2θ = 1 - 2(1/9) = 1 - 2/9 = 7/9

c. tan 2theta

First find tantheta.

tanθ = sinθ/cosθ; tanθ = (1/3) / (2sqrt(2)/3) = 1/3 * 3/(2sqrt(2)) = sqrt(2)/4; tan2θ = 2tanθ/(1 - (tanθ)^2); tan2θ = [2(sqrt(2)/4)] / [1 - (sqrt(2)/4)^2] = [sqrt(2)/2] / [1 - 2/16] = [sqrt(2)/2] / [14/16] = 4sqrt(2)/7