Question

When working in decimal radians, how do you find ${{\sin }^{-1}}\left( -\dfrac{1}{6} \right)$

Answer 1

Hint: In this question we will use the sine inverse of a negative value, we will first check whether the value lies in the domain of the sine inverse function and then using a calculator we will find the solution using decimal radians.

Complete step-by-step solution:
We have the given expression as ${{\sin }^{-1}}\left( -\dfrac{1}{6} \right)$
Now we that ${{\sin }^{-1}}(-x)=-{{\sin }^{-1}}x$ therefore, on using this formula on the given term, we get:
$\Rightarrow -{{\sin }^{-1}}\left( \dfrac{1}{6} \right)$
Now the domain of the sine inverse function is $[-1,1]$ and since the value $\dfrac{1}{6}\approx 0.1666$, it lies in the domain of the sine inverse function. On substituting the decimal value in the inverse function, we get:
$\Rightarrow -{{\sin }^{-1}}\left( 0.1666 \right)$
Now on taking the value of sine inverse in the form of radians, we get:
$\Rightarrow -{{\sin }^{-1}}\left( 0.1666 \right)=-0.167448079~rad$ , which is the required solution.

Note: Basic trigonometric formulas should be remembered to solve these types of sums. It is to be remembered which trigonometric functions are positive and negative in what quadrants.
When you add ${{180}^{\circ }}$ to any angle, its position on the graph reverses, and whenever you add ${{360}^{\circ }}$ to any angle, it reaches the same point after a complete rotation. There also exist half angle formulas which are an addition to the general angle’s addition-subtraction formulas.
In this question all the value of the inverse is given to us in radians, the symbol of degrees is $\circ $ and angle can also be represented in radians where $\pi $ is used which is equal to $180$ degrees. The other identity formula for cosine and tangent should be remembered too and whenever there is a trigonometric proof required, all the terms in the equation should be converted to the basic trigonometric identities of sine and cosine.