If ( )/(2) x (3 )/(4), then ^-1((12)/(13) x+(5)/(13) x) is equal to

Question

If $\frac{\pi }{2}\le x\le \frac{3\pi }{4}$, then ${\cos }^{-1}(\frac{12}{13}\cos x+\frac{5}{13}\sin x)$ is equal to

Accepted Answer

$$\begin{aligned}
& \frac{\pi}{2} \leq x \leq \frac{3 \pi}{4} \
& \cos ^{-1}\left(\frac{12}{13} \cos x+\frac{5}{12} \sin x\right) \
& \cos ^{-1}(\cos x \cos \alpha+\sin x \sin \alpha) \
& \cos ^{-1}(\cos (x-\alpha)) \
& \Rightarrow x-\alpha \text { because } x-\alpha \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \
& \Rightarrow x-\tan ^{-1} \frac{5}{12}
\end{aligned}$$

Solutions · 1