Inverse Sine

Every trigonometric function, be it Sine, Cosine, Tangent, Cotangent, Secant or Cosecant has an inverse of it, though in a restricted domain. To understand inverse sine out of other Inverse trigonometric functions, we need to study Sine first out of the other six trigonometric functions.

Sine Function

Sin (the sine function) takes an angle θ and produces a ratio. Sine is the ratio of the side opposite the angle θ to the Hypotenuse.

Sin θ = Opposite / Hypotenuse

Inverse Sine

Inverse Sine Function

  • Sin-1 (the inverse sine) takes the ratio Opposite/ Hypotenuse and produces angle θ. It is also written as arcsin or asine.

Arcsine:

Question: In a triangle ABC, AB= 4.9m, BC=4.0 m, CA=2.8 m and angle B = 35°.

Inverse Sine Example

Solution:

  • Sin 35° = Opposite / Hypotenuse
  • Sin 35° = 2.8 / 4.9
  • Sin 35° =0.57°

So, Sin-1 (Opposite / Hypotenuse) = 35°

Sin-1 (0.57)=35°

Let’s recall the Sin function again to understand Inverse sine.

  • Sin θ: length of the side opp to angle θ divided by the length of the hypotenuse.

OR

  • Sin θ = Opposite / Hypotenuse

Inverse Sine Formula

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° and the cable’s length is 40 m.

  •             Sin 39° = Opposite / Hypotenuse
  •             Sin 39° =  d / 40
  •             d = Sin 39° × 40
  •             d = 0.6293 ×  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 = Opposite / Hypotenuse, has?

Sin inverse is denoted by Sin-1 or arcsin.

Solution:

Let’s take the measurement from above example only.

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

We want to find “a”

Step 1: Find the sin a°

  • Sin a°  = Opposite / Hypotenuse
  • Sin a° = 25.17 / 40
  • Sin a° = 0.6293

Now, for which angle sin a°  = 0.6293

Let’s find it out with Inverse sin:

a ° = Sin-1 / (0.6293)

a °  = 38.1°

Did you know: Sin and Sin-1 are vice-versa.

Example: Sin 30°  = 0.5 and Sin-1 0.5 = 30°

Inverse Sine Graph

Arcsine trigonometric function is the sine function is shown as sin-1 a and is shown by the below graph.

Inverse Sign Derivative:

  • f Sinθ = θ
  • f’(Sin θ)(Cos θ) = 1
  • f’(Sin θ)= 1 / cos θ
  • A = sin θ = Cos θ = √(1-x2)
  • f’(x)=1 / √(1-x2)
  • d/dx Sinx= 1 / √(1-x2)

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Practise This Question

In the options given below, there are a few terms picked from an algebraic expression in x and y.
Which of them is a constant?