Question

Sin -1 [cos(43pi/5)] find the value

Answer 1

=>sin-1 [cos (8pi+3pi/5)]
=>sin-1 [cos (3pi/5)]
=> sin-1 [cos (pi/2+pi/10)]
=> sin-1 [-sin(pi/10)]
=>-sin-1 [sin(pi/10)]
=>-pi/10

Answer 2

$\bold{\text{The value of}} \ \bold{\sin ^{-1}\left(\cos \left(\frac{43 \pi}{5}\right)\right) \text { is }-\frac{\pi}{10}}$

To find:

Sin inverse [cos(43Π/5)] = ?

Step-by-step explanation:

By using trigonometric identities, cos trigonometric function is positive in first and fourth quadrants in coordinate axes.

$\sin ^{-1}\left(\cos \left(\frac{43 \pi}{5}\right)\right)=\sin ^{-1}\left(\cos \left(\frac{40 \pi}{5}+\frac{3 \pi}{5}\right)\right)$

$=\sin ^{-1}\left(\cos \left(8 \pi+\frac{3 \pi}{5}\right)\right)$

As $(8 \pi+\theta)$ is in First quadrant, cos is positive

$=\sin ^{-1}\left(\cos \left(\frac{3 \pi}{5}\right)\right)$

$=\sin ^{-1}\left(\cos \left(\pi-\left(\frac{2 \pi}{5}\right)\right)\right)$

As $( \pi-\theta)$ is in Second quadrant, cos is negative

$=\sin ^{-1}\left(-\cos \left(\frac{2 \pi}{5}\right)\right)$

$=-\left(\sin ^{-1}\left(\cos \left(\frac{2 \pi}{5}\right)\right)\right)$

We know that,

$\sin ^{-1}(x)+\cos ^{-1}(x)=\frac{\pi}{2}$

On substituting the above trigonometric identity, we get,

$\Rightarrow \sin ^{-1}\left(\cos \left(\frac{43 \pi}{5}\right)\right)=-\left(\frac{\pi}{2}-\cos ^{-1}\left(\cos \left(\frac{2 \pi}{5}\right)\right)\right)$

On solving, we get,

$=-\frac{\pi}{2}+\frac{2 \pi}{5}$

$=\frac{-5 \pi+4 \pi}{10}$

$\therefore \sin ^{-1}\left(\cos \left(\frac{43 \pi}{5}\right)\right)=-\frac{\pi}{10}$