Inverse Trigonometry Formula

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What is an Inverse Trigonometric Function?

In geometry, the part that tells us about the relationships existing between the angles and sides of a right-angled triangle is known as trigonometry. It has formulas and identities that offer great help in mathematical and scientific calculations. Along with that trigonometry also has functions and ratios such as sin, cos, and tan. In the same way, we can answer the question of what is an inverse trigonometric function? 

Well, there are inverse trigonometry concepts and functions that are useful. Inverse trigonometry formulas can help you solve any related questions. Just as addition is an inverse of subtraction and multiplication is an inverse of division, in the same way, inverse functions in an inverse trigonometric function. We can call it by different names such as anti-trigonometric functions, arcus functions, and cyclometric functions. The inverse trigonometric functions are as popular as anti trigonometric functions. The inverse functions have the same name as functions but with a prefix “arc” so the inverse of sine will be arcsine, the inverse of cosine will be arccosine, and tangent will be arctangent. So now when next time someone asks you what is an inverse trigonometric function?  You have a lot to say.  

y=sin-1x

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Inverse Trigonometric Functions Formulas

Inverse trigonometric functions formula helps the students to solve the toughest problem easily, all thanks to inverse trigonometry formula. Some of the inverse trigonometric functions formulas are:

  1. sin-1(x) = - sin-1x

  2. cos-1(x) = π - cos-1x

  3. sin-1(x) + cos-1x = π/2

  4. tan-1(x)+tan-1(y) = π + tan-1\[(\frac{x+y}{1-xy})\]

  5. 2sin-1(x) = sin-1(2x\[\sqrt{1-x^2}\])

  6. 3sin-1(x) = sin-1(3x - 4x3)

  7. sin-1x + sin-1y = sin-1( x\[\sqrt{1-y^2}\] + y\[\sqrt{1-x^2}\]), if x and y ≥ 0 and x2+ y2  ≤ 1

  8. cos-1x + cos-1y = cos-1(xy - \[\sqrt{1-x^2}\] + y\[\sqrt{1-y^2}\]), if x and y ≥ 0 and x2 + y2 ≤ 1

So these were some of the inverse trigonometric functions formulas that you can use while solving trigonometric problems


Table of Inverse Trigonometric Functions

Function Name

Notation

Definition

Domain of  x

Range

Arcsine or inverse sine

y = sin-1(x)

x=sin y

−1 ≤ x ≤ 1

− π/2 ≤ y ≤ π/2

-90°≤ y ≤ 90°

Arccosine or inverse cosine

y=cos-1(x)

x=cos y

−1 ≤ x ≤ 1

0 ≤ y ≤ π

0° ≤ y ≤ 180°

Arctangent or

Inverse tangent

y=tan-1(x)

x=tan y

For all real numbers

− π/2 < y < π/2

-90°< y < 90°

Arccotangent or

Inverse Cot

y=cot-1(x)

x=cot y

For all real numbers

0 < y < π

0° < y < 180°

Arcsecant or

Inverse Secant

y = sec-1(x)

x=sec y

x ≤ −1 or 1 ≤ x

0≤y<π/2 or π/2<y≤π

0°≤y<90° or 90°<y≤180°

Arccosecant

y=cosec-1(x)

x=cosec y

x ≤ −1 or 1 ≤ x

−π/2≤y<0 or 0<y≤π/2

−90°≤y<0°or 0°<y≤90°


Fun Facts

  • Hipparchus, the father of trigonometry compiled the first trigonometry table

  • Inverse trigonometric functions were actually introduced early in 1700x by Daniel Bernoulli. 


Solved Examples 

Example 1) Find the value of tan-1(tan 9π/ 8 )

Solution 1)  tan-1(tan9π/8)       

= tan-1tan ( π + π/8)

 = tan-1 (tan(π/ 8))

=π / 8

Example 2) Find sin (cos-13/5)

Solution 2) Suppose that, cos-1 3/5 = x

So, cos x = 3/5

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

So, sin x = \[\sqrt{1 - 9/25}\] = 4/5

This implies, sin x = sin (cos-1 3/5) = ⅘

Example 3) Prove the equation “Sin-1 (-x) = - Sin-1 (x), x ϵ (-1, 1)”

Solution 3) 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)

Example 4) Prove - Cos-1 (4x3 -3 x) =3 Cos-1 x , ½ ≤ x ≤ 1

Solution 4)  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

Example 5) Differentiate y = \[\frac{1}{sin^{-1}x}\]

Solution 5) Using the inverse trigonometric functions formulas along with the chain rule

= \[\frac{dy}{dx}\] = \[\frac{d}{dx}\](sin-1x)-1

= -(sin-1x)-2 \[\frac{d}{dx}\](sin-1x)

= -\[\frac{1}{(sin^{-1}x)^2\sqrt{(1-x^{2})}}\]

FAQ (Frequently Asked Questions)

Question 1) What are the applications of Inverse Trigonometric Functions?

Answer 1) The inverse trigonometric formula’s major role is to help us in finding out the unknown measurement of an angle of a right angle triangle when any of its two sides are provided. We use the trigonometric function particularly on the basis of which sides are known to us. In other words, if the measurement of the side of the hypotenuse and the side opposite to the angle ϴ are known to us then we use an inverse sine function. In the same way, if we are provided with the measurement of the adjacent side and the opposite side then we use an inverse tangent function for the determination of a right-angle triangle. The inverse trigonometric function extends its hand even to the field of engineering, physics, geometry, and navigation. An inverse trigonometric function can be determined by two methods. The first is to use the trigonometric ratio table and the second is to use scientific calculators.

Question 2) What are Trigonometric Functions?

Answer 2) Trigonometry is the science of measuring triangles. We can refer to trigonometric functions as the functions of an angle of a triangle. In other words, it is these trig functions that define the relationship that exists between the angles and sides of a triangle. 

There are six main trigonometric functions that are given below:

  1. Sine (sin)

  2. Cosine (cos)

  3. Tangent (tan)

  4. Secant (sec)

  5. Cosecant (cosec)

  6. Cotangent (cot)

We use these functions to relate the angles and the sides of a right-angled triangle. Trigonometric functions are important when we are studying triangles.