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 can find $\sin 2\theta$ using the identity $\sin^2 2\theta = 1 - \cos^2 2\theta$. Substituting the given value, we get $\sin^2 2\theta = 1 - \left(-\frac{24}{25}\right)^2 = 1 - \frac{576}{625} = \frac{49}{625}$, so $\sin 2\theta = \pm \frac{7}{25}$. Since $\frac{\pi}{2} < 2\theta < \pi$, $\sin 2\theta$ is positive, so $\sin 2\theta = \frac{7}{25}$. Using the double angle formula for sine, $\sin 2\theta = 2\sin\theta\cos\theta$, and knowing $\cos 2\theta = 2\cos^2\theta - 1$, we can find $\sin\theta$ and $\cos\theta$. However, we only need $\sin\theta$, and given the constraints, we can deduce that $\sin\theta$ must be positive in the given interval. Therefore, using the Pythagorean identity, $\sin\theta = \sqrt{1 - \cos^2\theta}$, and from $\cos 2\theta$, we can indirectly find $\cos\theta$ and thus $\sin\theta$. The correct answer, considering the positive square root and the given interval, is $\sin\theta = \frac{7\sqrt{2}}{10}$, which corresponds to choice C.

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

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