Question

The principal value of ${{\sin }^{-1}}(\sin 10)$ is
(a) $10$
(b) $10-3\pi$
(c) $3\pi -10$
(d) None of these

Answer 1

Hint: We will first check if 10 radians will lie in the range of sin x or not. If not then we will find the angle in terms of $\pi$ which will lie in the range between $-\dfrac{\pi }{2}$ and $\dfrac{\pi }{2}$. Using this information we will get the answer.
Complete step-by-step answer:
Before proceeding with the question, we should understand the definition of principal value.
The principal value of ${{\sin }^{-1}}x$ for x>0, is the length of the arc of a unit circle centred at the origin which subtends an angle at the centre whose sine is x. For this reason it is also denoted by arcsin x.
Now let us assume that ${{\sin }^{-1}}(\sin 10)=x.....(1)$
Also we know that the range is between $-\dfrac{\pi }{2}$ and $\dfrac{\pi }{2}$. So now rearranging the terms in equation (1) we get,
$\Rightarrow \sin x=\sin 10........(2)$
Here the angle given is 10 radians and it does not lie between $-\dfrac{\pi }{2}$ and $\dfrac{\pi }{2}$.
But, $3\pi -10$ lies between $-\dfrac{\pi }{2}$ and $\dfrac{\pi }{2}$.
Also we know that $\sin (3\pi -10)=\sin 10$ and hence substituting this in equation (2) we get,
$\Rightarrow \sin x=\sin (3\pi -10)........(3)$
Now solving for x in equation (3) by cancelling both sides we get,
$\Rightarrow x=3\pi -10$
Hence the correct answer is option (c).

Note: We need to be clear about what the principal value in trigonometry means. Also we need to understand the difference between radians and degrees. And we need to remember the range and domains of different inverse trigonometric functions.