Question

How do you find inverse trigonometric functions without a calculator?

Answer 1

Hint : First we will specify the basic requirements that are needed initially in order to solve the problem. Then we will mention all the trigonometric terms and how they are interrelated. Then we mention all the equations.

Complete step-by-step answer :
Now in these kinds of questions, the easiest way to work with inverse trigonometric functions is to have a chart handy with the exact values of the functions. When you constantly solve trigonometric questions, you soon have the basic angles and their particular function values memorised.
We also know that sine and its reciprocal cosecant are positive in the first quadrant and second quadrant. The cosine and its reciprocal secant are positive in the first quadrant and fourth quadrant. The tangent and its reciprocal cotangent are positive in the first and third quadrant. Now, with these basics you can come up with function values rather quickly and efficiently without using calculators or charts.
We can sum up the above mentioned equations as,

$$\cos ine = \dfrac{1}{{\sin }} \\\ \sec = \dfrac{1}{{\cos }} \\\ \cot = \dfrac{1}{{\tan }} \;$$

If $\tan x = \theta$ then $x= tan^{-1}\theta$ this is how we find inverse trigonometric functions without using a calculator.

Note : While choosing the side to solve, always choose the side where you can directly apply the trigonometric identities. Also, remember the trigonometric identities ${\sin ^2}x + {\cos ^2}x = 1$ and $\cos 2x = 2{\cos ^2}x - 1$. While opening the brackets make sure you are opening the brackets properly with their respective signs. Also remember that $\tan x = \,\dfrac{{\sin x}}{{\cos x}}$.