Solve the problem.
-If $\cos \theta = - \frac { 5 } { 13 }$, and $\frac { \pi } { 2 }

Question

Solve the problem.
-If $\cos \theta = - \frac { 5 } { 13 }$, and $\frac { \pi } { 2 } < \theta < \pi$, then find $\tan 2 \theta$.
A) $\frac { 120 } { 119 }$

B) $\frac { 169 } { 119 }$

C) $\frac { 169 } { 120 }$

D) $\frac { 119 } { 120 }$

Quizplus · Accepted Answer

Given $\cos \theta = -\frac{5}{13}$, and $\theta$ is in the second quadrant ($\frac{\pi}{2} < \theta < \pi$), we can find $\sin \theta$ using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Since $\cos \theta = -\frac{5}{13}$, we have $\sin \theta = \pm\frac{12}{13}$. In the second quadrant, $\sin \theta > 0$, so $\sin \theta = \frac{12}{13}$. The formula for $\tan 2\theta$ is $\frac{2\tan \theta}{1 - \tan^2 \theta}$. Since $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{12/13}{-5/13} = -\frac{12}{5}$, we substitute into the formula to get $\tan 2\theta = \frac{2(-\frac{12}{5})}{1 - (-\frac{12}{5})^2} = \frac{-24/5}{1 - 144/25} = \frac{-24/5}{-119/25} = \frac{120}{119}$.

Solve the Problem cos⁡θ=−513\cos \theta = - \frac { 5 } { 13 }cosθ=−135​

Solve the Problem $\cos \theta = - \frac { 5 } { 13 }$