Given sin theta = (-12/13), and cos theta > 0, how do you find the exact values of sin(2theta) and cos(2theta)?

Mar 5, 2017

Answer:

$\sin 2 \theta = - \frac{120}{169}$ and $\cos 2 \theta = - \frac{119}{169}$

Explanation:

As $\sin \theta = - \frac{12}{13}$ i.e. negative and $\cos \theta$ is positive, $\theta$ lies in $Q 4$ i.e. $\frac{3 \pi}{2} < \theta < 2 \pi$

$\cos \theta = \sqrt{1 - {\left(- \frac{12}{13}\right)}^{2}} - \sqrt{1 - \frac{144}{169}} = \sqrt{\frac{25}{169}} = \frac{5}{13}$

Hence, $\sin 2 \theta = 2 \sin \theta \cos \theta = 2 \times \left(- \frac{12}{13}\right) \times \frac{5}{13} = - \frac{120}{169}$

and $\cos 2 \theta = {\cos}^{\theta} - {\sin}^{2} \theta = \frac{25}{169} - \frac{144}{169} = - \frac{119}{169}$