Introduction

The equation sin q = x means that q is an angle whose sine is x. It is often convenient to express this inversely by writing q = sin-1 x. Thus, the symbol sin-1 x denotes an angle whose sine is x. Hence, sin-1 x is an angle, whereas sin q is a number. The two relations sin q = x and q = sin-1x is usually read as "sine inverse x". Sometimes it is also denoted by arc sin x.

Note: sin-1 x must not be confused with (sin x)-1, i.e., (1/sin x)

We know that if q be any one of the value whose sine is equal to x, then the sine of all the angles given by sin-1 x = np + (-1)n sin-1 x are equal to x. Hence, sin-1 x has got an infinite number of values and as such sin-1 x is a multiple valued function.

Similarly, the general values of cos-1 x = 2np ± cos-1x and tan-1x
= np + tan-1x.

The smallest numerical value, either positive or negative of q is called the principal value of sin-1x. Thus the principal value of sin-1(1/2)is 30o. If corresponding to the same ratio, there are two numerically equal angles, one positive and other negative, it is customary to take the positive angle as the cos-1 ½ is 60o, and not (- 60o) although cos (- 60o) = ½.

Similar significance and all properties as those of sin-1x. These expressions are called inverse circular functions.