Question

How do you evaluate $\arcsin ( - 1)$ without a calculator?

Answer 1

Hint : First we will evaluate the right-hand of the equation and then further the left-hand side of the equation. We will now consider $x = {\sin ^{ - 1}}\left( { - 1} \right)$ and then we will further this expression to $\sin x$ form and hence evaluate the value of the term.

Complete step-by-step answer :
We will start off by considering $x = {\sin ^{ - 1}}\left( { - 1} \right)$ .
Hence, the equation will become,
$x = {\sin ^{ - 1}}\left( { - 1} \right) \\\ \sin x = - 1 \;$
Now we know that the value of $\sin \left( {\dfrac{\pi }{2}} \right) = 1$ and $\sin \left( {\dfrac{{ - \pi }}{2}} \right) = - 1$ .
So now here, the value will be,
$\arcsin ( - 1) = \dfrac{{ - \pi }}{2}$
So, the correct answer is “ $\dfrac{{ - \pi }}{2}$ ”.

Note : Inverse trigonometric functions are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant and cosecant functions. They are also termed as arcus functions, anti trigonometric functions or cyclometric functions. If an inverse function exists for a given function $f$, then it is unique. This follows since the inverse function must be the converse relation, which is completely determined by $f$. There is a symmetry between a function and its inverse.
While applying the double angle identities, first choose the identity according to the terms you have then choose the terms from the expression involving which you are using the double angle identities. While modifying any identity make sure that when you back trace the identity, you get the same original identity.