Inverse Sine

To understand inverse sine, we need to study Sine first.

What is Sine?

  • Sin (the sine function) takes an angle θ and produces a ratio

Sin \(\theta\)= \(\frac{Opposite}{Hypotenuse}\)

Inverse Sine

Inverse Sine Function

  • \(Sin^{-1}\)/, (the inverse sine) takes the ratio \(\frac{Opposite}{Hypotenuse}\)/, and produces \(\angle \theta\).

Example of Inverse sine:

Question: In a triangle ABC, AB= 4.9m, BC=4.0 m, CA=2.8 m and \(\angle B\) = 35 \(^{\circ}\).

Inverse Sine Example




Sin35\(^{\circ}\)=0.57 \(^{\circ}\)

So, \(Sin^{-1}\)\(\frac{Opposite}{Hypotenuse}\)=35\(^{\circ}\)


We need to understand the Sin function first to understand Inverse sine.

Sin\(\angle \theta\): length of the side opp to angle θ /

Divided by the length of the hypotenuse.



For Example:

  • If we want to find the depth(d) of the seabed from the bottom of the ship and the following two parameters are given:
  • The angle which the cable makes with the seabed.
  • The cable’s length.

The Sine function will help to find the distance/depth d of the ship from the sea bed by the following method:

Step 1: If the angle is 39\(^{\circ}\)

and the cable’s length is 40 m.

Sin 39\(^{\circ}\) = \(\frac{Opposite}{Hypotenuse}\)

Sin 39\(^{\circ}\) = \(\frac{d}{40}\)

d = Sin 39\(^{\circ}\) \(\times\)40

d = 0.6293 \(\times\) 40

d = 25.172 cm

So the depth d is 25.17 cm.

Now, if the angle is not given and we want to calculate it, then we use the Inverse function and the question will be asked in the following way:

So, the question is asked in the following manner?

Question: What is the\(\angle\) Sin =\(\frac{Opposite}{Hypotenuse}\)/, has ?

Sin inverse is denoted by \(Sin^{-1}\) or arcsin.


Let’s take the measurement from above example only.

  • Distance d = 25.17 cm
  • Cable’s length = 40 cm.

We want to find the \(\angle\)a

Step 1: Find the sin a\(^{\circ}\)

  • Sin a\(^{\circ}\) = \(\frac{Opposite}{Hypotenuse}\)
  • Sin a\(^{\circ}\)= \(\frac{25.17}{40}\)
  • Sin a\(^{\circ}\)=0.6293

Now, for which angle sin a\(^{\circ}\) = 0.6293

Let’s find it out with Inverse sin:

a \(^{\circ}\)= \(Sin^{-1}\)/(0.6293)

a \(^{\circ}\) = 38.1 \(^{\circ}\)

Did you know: Sin and \(Sin^{-1}\) are vice-versa.

Example: Sin 30 \(^{\circ}\) = 0.5 and \(Sin^{-1}\) 0.5 = 30 \(^{\circ}\)

Derivative of Inverse Sign:

f(Sin \(\theta\)= \(\theta\)

f’(Sin\(\theta\))Cos \(\theta\) = 1

f’(Sin \(theta\))= \(\frac{1}{cos\theta }\)

A = sin \(theta\)) = Cos \(\theta\)= \(\sqrt{1-x^{2}}\)



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Practise This Question

The sum of first n consecutive odd numbers is equal to n2.