Question

How do you use a calculator to evaluate \[{{\sin }^{-1}}0.75\] in both radians and degree?

Answer 1

Hint: We explain the function \[{{\sin }^{-1}}0.75\]. We express the inverse function of sin in the form of $ \arcsin \left( x \right)={{\sin }^{-1}}x $ . We need to solve the equation in the calculator for both radians and degree. We explain the process of how to solve the problem in the calculator.

Complete step by step answer:
First, we find the general solution of the equation \[{{\sin }^{-1}}0.75\]. The calculator settings usually remain in degree mode. There has to be a sign on the display board which is either ‘D’ or ‘R’ which tells us that it is either in degree or radian respectively.
Now if it’s in degree then we need to first press the button which is for \[{{\sin }^{-1}}x\] function. The main function in the board is \[\sin x\]. The same button is used along with ‘Shift’ gives the command for \[{{\sin }^{-1}}x\]. After pressing that button, we put the value for \[0.75\]. We can either use it as a decimal or we can also use the fraction which is $ 0.75=\dfrac{75}{100}=\dfrac{3}{4} $.
The rounded off value of \[{{\sin }^{-1}}0.75\] in degree will be \[{{\sin }^{-1}}0.75={{48.6}^{\circ }}\].
Now if we want to find it in radian, we press the key with ‘mode’ written over it. The first thing that comes on screen is a choice of a radian, degree, and many others. We click the required numeric value mentioned on the screen to convert into radian mode and do the same process to find the radian value which will be \[{{\sin }^{-1}}0.75=0.85\]. (approx.)
Let’s assume the degree value is x which means \[{{\sin }^{-1}}0.75=x\]. We can convert it into radian using the relation where 1 degree is equal to $ \dfrac{\pi }{180} $ radian. This gives x degree is equal to $ \dfrac{\pi x}{180} $ radian. We evaluate the value of \[{{\sin }^{-1}}0.75\] in both radians and degree as x degree and $ \dfrac{\pi x}{180} $ radian where \[\sin x=0.75\].

Note:
The domain for the degree value in the calculator and also in the primary solution for the inverse of ratio sin is $ -\dfrac{\pi }{2}\le x\le \dfrac{\pi }{2} $. Instead of using relation to get the radian value, we can also directly use the calculator to find the radian value by just changing the setting from degree to radian.