Inverse trigonometric functions are easy to understand

The inverse trigonometric functions are defined for all the trigonometric functions. The standard notations for trigonometry inverse functions are sin-1y, cos-1y, tan-1y, cot-1y, sec-1y and cosec-1y. The values of all this inverse trig functions can be found out by knowing the trigonometric ratio.

Study of basic inverse trigonometric identities:

The inverse trigonometric identities given below are extremely beneficial for solving inverse trigonometric functions. Its advised that this identities should be studied with great attention. The formulas of inverse trigonometric functions are given below.

  • sin-1(-y) = -sin-1y

  • sin-1(1/y) = cosec-1y and cos-1(1/y) = sec-1y

  • tan-1(1/y) = cot-1y

  • sin-1y + cos-1y = π/2

  • tan-1y + cot-1y = π/2

  • sec-1y + cosec-1y = π/2

  • tan-1y + tan-1z = tan-1((x+y) / (1-xy))

We have seen the formulas for inverse trigonometric function. But study of this formulas is not enough. The properties of inverse trigonometric functions are required to solve the inverse trig functions problems correctly.

  • sin(sin-1x) = x for x bound between [-1,1]. Also sin-1(sin y) = y for y bound between [-π/2,π/2].

  • Similarly tan(tan-1x) = x and tan-1(tan y) = y for y bound between [-π/2,π/2].

Similar formulas can be written for all types of inverse functions. Now lets have a look at inverse trigonometric functions problems and try to analyze them.

(1) Evaluate sin-1(1):

We need to find the angle θ such that θ = sin-1(1) or sin θ = 1. We know that sin 90° = 1. Hence the answer is 90°.

(2) Prove that sin-1(2y√(1-y2)) = 2cos-1y:

Assume y = cosθ => cos-1y = θ. From the equation, sin-1(2y√(1-y2)) = sin-1(2cosθ√(1-cos2θ)). Now since 1-cos2θ= sin2θ, the answer is sin-1(2cosθsinθ) = sin-1(sin2θ) = 2θ = 2cos-1y. Thus the result is proved.

The inverse trig functions practice problems are left below to have some practice. This inverse trigonometric functions practice problems shouldn't be ignored and solved seriously so as to have vast knowledge of the topic.

(1) Find sin-1y for y = 0 and y = ½.

(2) Evaluate tan-1(√(y2-1)) to its simplest form.

Study of inverse trig function graphs:

The inverse trigonometric functions graphs are given to know how graphs are drawn for inverse functions. The inverse trigonometric functions graphs are given for sin-1x and cos-1x functions.
inverse trigonometric functions

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