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 $\frac { \pi } { 2 } < \theta < \pi$, we are in the second quadrant where cosine is negative, sine is positive, and $\tan \theta$ is negative. The Pythagorean identity gives us $\sin^2 \theta = 1 - \cos^2 \theta$, so $\sin \theta = \sqrt{1 - \left(-\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13}$. Since we're in the second quadrant, $\sin \theta$ remains positive. To find $\tan 2\theta$, we use the double angle formula: $\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$. First, find $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{12/13}{-5/13} = -\frac{12}{5}$.Substitute $\tan \theta = -\frac{12}{5}$ into the double angle formula: $\tan 2\theta = \frac{2(-\frac{12}{5})}{1 - (-\frac{12}{5})^2} = \frac{-\frac{24}{5}}{1 - \frac{144}{25}} = \frac{-\frac{24}{5}}{\frac{-119}{25}} = \frac{24}{5} \cdot \frac{25}{-119} = \frac{120}{-119} = \frac{120}{119}$.Therefore, the correct answer is $\frac{120}{119}$, which is choice A.

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

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