Question

If sin theta equals 12 divided by 13 and theta is in quadrant 2 cos20 equals? and cos theta equals?

Answer 1

Answers:

• cos(2theta) = -119/169
• cos(theta) = -5/13

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The given info is:

• sin(theta) = 12/13
• theta is in quadrant II

From that, we can use the pythagorean trig identity to find that cos(theta) = -5/13. Keep in mind that cosine is negative in quadrant II.

Now use the trig identity below to compute cos(2theta)

$\cos(2\theta) = \cos^2(\theta)-\sin^2(\theta)\\\\\cos(2\theta) = \left(\frac{-5}{13}\right)^2-\left(\frac{12}{13}\right)^2\\\\\cos(2\theta) = \frac{25}{169}-\frac{144}{169}\\\\\cos(2\theta) = \frac{25-144}{169}\\\\\cos(2\theta) = -\frac{119}{169}$

Other options you could use are these identities

$\cos(2\theta) = 2\cos^2(\theta)-1$

or

$\cos(2\theta) = 1-2\sin^2(\theta)$