Question

Given that sine theta sinθ= 13/85 and cosine theta <​0, determine the values of the sine and cosine functions for 2thetaθ.

Answer 1

Find $cos\theta$ first ($cos\theta < 0$) :

$$cos \theta = - \sqrt{1 - sin^2 \theta} = -\sqrt {1 - (13/85)^2} = -84/85$$

Find $sin 2 \theta$ :

$$sin 2 \theta = 2 sin \theta cos \theta = - 2 \cdot \frac{13}{85} \cdot \frac {84}{85} = - \frac{2184}{7225}$$

Find $cos 2 \theta$ :

$$cos 2 \theta = cos^2 \theta - sin^2 \theta = (-\frac{84}{85})^2 - (\frac{13}{85})^2 = \frac{6887}{7225}$$