Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\csc \theta = \frac { 17 } { 15 } , \frac { \pi } { 2 }

Question

Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\csc \theta = \frac { 17 } { 15 } , \frac { \pi } { 2 } < \theta < \pi$
Find $\cos ( 2 \theta )$.
A) $- \frac { 161 } { 289 }$

B) $\frac { 161 } { 289 }$

C) $\frac { 240 } { 289 }$

D) $- \frac { 240 } { 289 }$

Quizplus · Accepted Answer

Given $\csc \theta = \frac{17}{15}$, we can find $\sin \theta = \frac{15}{17}$. Since $\frac{\pi}{2} < \theta < \pi$, $\theta$ is in the second quadrant where $\sin \theta > 0$ and $\cos \theta < 0$. We can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find $\cos \theta$. $\sin \theta = \frac{15}{17}$ implies $\sin^2 \theta = \frac{225}{289}$. Thus, $\cos^2 \theta = 1 - \frac{225}{289} = \frac{64}{289}$. Since $\theta$ is in the second quadrant, $\cos \theta = -\frac{8}{17}$.To find $\cos(2\theta)$, we use the double angle formula: $\cos(2\theta) = 2\cos^2 \theta - 1$. Substituting $\cos \theta = -\frac{8}{17}$ gives $\cos(2\theta) = 2\left(-\frac{8}{17}\right)^2 - 1 = 2\left(\frac{64}{289}\right) - 1 = \frac{128}{289} - 1 = -\frac{161}{289}$.

Use the Information Given About the Angle θ,0≤θ≤2π\theta , 0 \leq \theta \leq 2 \piθ,0≤θ≤2π

Use the Information Given About the Angle $\theta , 0 \leq \theta \leq 2 \pi$