Inverse sine functions in trigonometry

The arcsin x notation indicates that we are talking about the angle whose sine function gives x. Sometimes inverse sine function is also symbolized as sin-1x. According to the inverse sine function definition, the inverse function of the trigonometric ratio sine taken over a specified domain gives the inverse sine functions.

The inverse trigonometric functions are often called cyclometric functions. The inverse function of sine finds wide range of application in the fields of engineering and several physical theories. Lets move on to the graph to understand them better.

Graph of inverse sine function

The inverse sine function graph is as given below. If we recall the graph of sine function, it is evident that its just a reflection of the sine function.
inverse sine function graph

It should be noted that y = sin-1x is different from y = 1/sinx. Instead it implies that x = siny. The domain of this inverse sine function is from -1 to 1 while the range is between (-π/2, π/2). This functions can be used for solving derivative and integration problems also. Several derivative and integration formulas involving inverse sine functions can be derived for use in problems. Now let study some inverse sine function examples and try to gain more out of it.

(1) Find the inverse sine value of z, where z = sin(0.765):

We have z = sin(0.765). Lets take inverse on both the sides. arcsin(z) = arcsin(sin 0.765). This implies that arcsin(z) = 0.765 or the inverse sine value of z is 0.765.

(2) Find the value of sin-1(1/2):

We know that sin 30° = ½. Taking the inverse on both sides, sin-1(sin 30°) = sin-1(1/2). But since sin-1 and sin are both complementary, they gets canceled. Hence sin-1(1/2) = 30.

(3) Evaluate sin-1(1):

Again, we have sin(90°) = 1. So taking the inverse on both sides and solving it yields the answer as 90. Thus sin-1(1) is equal to 90.

Thus we had a brief on inverse sine functions along with its graph, important properties and examples. This information is sufficient for the study of inverse trig sine functions.

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