Use a double-angle formula to find the exact value of cos2u when $$\sin u = \frac { 7 } { 25 } \text {, where } \frac { \pi } { 2 }

Question

Use a double-angle formula to find the exact value of cos2u when  $$\sin u = \frac { 7 } { 25 } \text {, where } \frac { \pi } { 2 } < u < \pi$$
A) $\cos 2 u = \frac { 527 } { 625 }$
B)  $\cos 2 u = - \frac { 1152 } { 625 }$
C)  $\cos 2 u = \frac { 336 } { 625 }$
D)  $\cos 2 u = \frac { 168 } { 625 }$
E)  $\cos 2 u = - \frac { 478 } { 625 }$

Quizplus · Accepted Answer

Given $\sin u = \frac{7}{25}$, we can find $\cos u$ using the Pythagorean identity $\sin^2 u + \cos^2 u = 1$. This gives $\cos u = \pm\frac{24}{25}$. Since $\frac{\pi}{2} < u < \pi$, $\cos u$ is negative in the second quadrant, so $\cos u = -\frac{24}{25}$. Using the double-angle formula for cosine, $\cos 2u = 2\cos^2 u - 1$, we get $\cos 2u = 2\left(-\frac{24}{25}\right)^2 - 1 = 2\left(\frac{576}{625}\right) - 1 = \frac{1152}{625} - \frac{625}{625} = \frac{527}{625}$.

Use a Double-Angle Formula to Find the Exact Value of Cos2u