Important Properties of Inverse Trigonometric Functions

Inverse Trigonometric Functions formulas

Inverse trigonometric functions are identified as the inverse functions of some basic trigonometric functions such as sine, cosine, tangent, secant, cosecant, and cotangent functions. Inverse trigonometric functions are also known as anti trigonometric functions, arcus functions, and cyclometric functions. These inverse trigonometric functions formulas enable us to find out any angles with any of the trigonometry ratios. These formulas are derived from the properties of trigonometric functions.Through this article, we learn about inverse trigonometry concepts, properties of inverse trigonometric functions, inverse trigonometric functions problems etc.

Inverse Trigonometric Functions

Inverse trigonometric functions also termed as “Arc functions” or anti trigonometric functions are the inverse functions of some basic trigonometric functions. They are used to find out the unknown values of angles of a right- angle triangle with any of the trigonometry ratios. We know that trigonometric functions are usually applied to the right angle triangle. They are widely used in the field of engineering, physics, geometry and navigation. The six trigonometric functions are used to find out the angle measurement of a right angle triangle when the measurement of  two sides of the triangle are known.


There are six trigonometric functions for each of the trigonometry ratios. The inverse of those six trigonometric functions are:

  • Arcsine

  • Arccosine

  • Arctangent

  • Arcsecant

  • Arcosecant

  • Arccontangent

Properties of Inverse Trigonometric Functions Properties

There are some basic properties of inverse trigonometric functions that are needed not merely to solve problems but also help to understand the concept of inverse trigonometric functions in a precise manner. As we know, inverse trigonometric functions also termed as “Arc functions” help us to learn how to find out the unknown values of angle through the trigonometry ratios.

Considering the trigonometry ratios, we will study here some important properties of inverse trigonometric functions

Important Properties of Inverse Trigonometric Functions 

Here, you can examine some important properties of inverse trigonometric functions.


Property 1

  1. Sin-1 (1/a) = cosec -1 a, a 1 or a -1

  2. Cos-1 (1/a) = sec -1 a, a 1 or a -1

  3. Tan-1 (1/a) = cot -1 a, a > 0

Property 2

  1. Sin-1 (-a)= - sin-1 (a), a ϵ (-1, 1)

  2. Tan-1 (-a) = - tan-1 (a), a ϵ R

  3. Cosec -1 (-a) = - cosec -1 (a), | a |  1

Property 3

  1. Cos-1 (-a) = π- cos -1 a, a ϵ (-1, 1)

  2. Sec-1 (-a) = π-sec -1 a | a | ≥1

  3. Cot-1 (-a) = π -cot -1 a, a ϵ R

Property 4

  1. Sin-1a + Cos-1 a = π/2, a ϵ (-1, 1)

  2. Tan-1 a + Cot-1 a = π/2, a ϵ R

  3. Cosec-1 a+ Sec -1 a = π/2, | a | 1

 Property 5

  1. Tan-1 a -Tan-1b= Tan-1 ((a+ b) / (1-ab)), ab <1

  2. Tan-1 a - Tan-1 b = Tan-1((a+ b) / (1+ab)),ab > -1

Property 6

  1. 2tan-1 a = tan-1 (2a (1 +a2)), | a | ≤ 1

  2. 2tan-1 a = cos-1 (2a (1- a2)/ (1+ a2)), a ≥ 0

  3. 2tan-1 a = tan-1(2a (1- a2)), -1 < a < 1

Inverse Trigonometric Functions Problems

Here, we will discuss inverse trigonometric functions problems

  1. Find the value of sin (cos-1 3/5)

Solution:

Let us consider cos-1 3/5 = x

So, cos x = 3/5

We know that, Sin x = \[\sqrt {1 - {{\cos }^2}x} \]

Accordingly, Sin x = \[\sin x = \sqrt {1 - \frac{9}{{25}}} \]

= 4/5

It implies that sin x, = Sin (cos-1 3/5) = 4/5

  1. What will be the value of x , if sin(Sin-1 1/5 + Cos-1 x)

Solution:

Sin-1 1/5 + cos-1 x = π/2

Sin-1 1/5 = π/2- cos-1 x= sin-1 x

X= 1/5

Solved Examples


  1. Prove the equation below:


“Sin-1 (-x) = - Sin-1 (x), x ϵ (-1, 1)”


Solution: Let Sin-1 (-x) = y

Then -x = sin y

x = - sin y

x =sin (-y)

sin-1 -x = arcsin ( sin(-y))

sin-1 -x = y

Hence, Sin-1 (-x) = - Sin-1 (x), x ϵ (-1, 1)


  1. Prove that  - Cos-1 (4x3 -3 x) =3 Cos-1 x , ½ x 1.


Solution: Let x = Cos ϴ

Where ϴ = Cos-1 (-x)

LHS = Cos-1 (-x) (4x3 -3x)

By substituting the value of x, we get

= Cos-1 (-x) (4 Cos3ϴ - 3 Cosϴ)

Accordingly, we get,

 Cos-1 (Cos 3ϴ)

= 3ϴ

By substituting the value of ϴ, we get

= 3 Cos-1 x

= RHS

Hence proved

Quiz Time

  1. What is the principal value of the expression Cos-1[Cos (-680°)] ?

a.       2 π/9

b.      -2 π/9

c.       π/9

d.      34 π/9

 2.     Tan-1 (sin- π/2) is equal to

a.  -1

b.  1

c.   π/2

d.  – π/4

Facts

  • Hipparchus is the father of trigonometry who compiled the first trigonometry table

  • Inverse trigonometric functions were introduced early in 1700x by Daniel Bernouli used A, sin for the inverse sine of a number

  • Euler wrote, “A t” for the inverse tangent in 1736.