[ ^ - 1( ( - 5)) + ^ - 1( (6)) - ^ - 1( (12))] is equal to :(The inverse trigonometric fun…

Question

$${\cos ^{ - 1}}(\cos ( - 5)) + {\sin ^{ - 1}}(\sin (6)) - {\tan ^{ - 1}}(\tan (12))$$ is equal to :(The inverse trigonometric functions take the principal values)

Accepted Answer

$\cos^{-1}(\cos(-5))=\cos^{-1}(\cos5)=2\pi-5$ (since $5\in(\pi,2\pi)$).

$\sin^{-1}(\sin6)=6-2\pi$ (since $6\in\left(\dfrac{3\pi}{2},2\pi\right)$).

$\tan^{-1}(\tan12)=12-4\pi$ (since $12-4\pi\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$).

$$(2\pi-5)+(6-2\pi)-(12-4\pi)=4\pi-11.$$

Solutions · 1