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Let us take a right angle triangle ABC, with right angle at B.

Sine of A is defined as BC / AC  = opposite side to angle A / hypotenuse

Sin A = \frac{BC}{AC}       example:  Sin 30 deg  = 1/2

We want to know the value of angle A, but we may not directly know it. Then we can calculate its sine value from other quantities in some way, and then find its value as:

 sin^{-1} 1/2 = 30\ deg\\

Thus  we define  sin^{-1} BC/AC\\ as equal to angle A.

Same way, we know  Cosine\ C = \frac{BC}{AC}
so we define Cos^{-1} \frac{BC}{AC} = angle C\\

Similarly we know the  tangent tan C = \frac{AB}{BC} \\
We want to find the angle C, when we are given AB and BC. So we define a function
 tan^{-1} \frac{AB}{BC}\ \ as\ angle\ C\\

So sine cosine, tangent of angles give ratio of sides in a right angle triangle. The inverse functions give the values of angles from ratio of sides.
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