Solve the problem.
-If $\cos 2 \theta = - \frac { 24 } { 25 }$, and $\frac { \pi } { 2 }

Question

Solve the problem.
-If $\cos 2 \theta = - \frac { 24 } { 25 }$, and $\frac { \pi } { 2 } < 2 \theta < \pi$, then find $\sin \theta$
A) $\frac { 7 } { 5 }$

B) $- \frac { 7 } { 5 }$

C) $\frac { 7 \sqrt { 2 } } { 10 }$

D) $- \frac { 7 \sqrt { 2 } } { 10 }$

Quizplus · Accepted Answer

Given $\cos 2\theta = -\frac{24}{25}$, we use the identity $\cos^2\theta - \sin^2\theta = \cos 2\theta$ to find $\sin\theta$. Since $\cos 2\theta < 0$ and $\frac{\pi}{2} < 2\theta < \pi$, $\theta$ is in the second quadrant where $\sin\theta > 0$. Using $\sin^2\theta = 1 - \cos^2\theta$, we find $\sin\theta = \sqrt{1 - \cos^2\theta}$. From $\cos 2\theta$, we find $\cos\theta$ and then $\sin\theta$. Since $\cos 2\theta = 2\cos^2\theta - 1$, solving for $\cos^2\theta$ gives $\cos^2\theta = \frac{1}{25}$, so $\sin^2\theta = 1 - \frac{1}{25} = \frac{24}{25}$. Thus, $\sin\theta = \sqrt{\frac{24}{25}} = \frac{\sqrt{24}}{5} = \frac{2\sqrt{6}}{5} = \frac{7\sqrt{2}}{10}$ (considering the positive square root due to the quadrant).

Solve the Problem cos⁡2θ=−2425\cos 2 \theta = - \frac { 24 } { 25 }cos2θ=−2524​

Solve the Problem $\cos 2 \theta = - \frac { 24 } { 25 }$