Inverse Sine

Inverse Sine is a trigonometric function which denotes the inverse of the sine function and is represented as – Sin-1. The formula for this function is simple to derive. Every trigonometric function, whether it is Sine, Cosine, Tangent, Cotangent, Secant or Cosecant has an inverse of it, though in a restricted domain. To understand the inverse of sine out of other Inverse trigonometric functions, we need to study Sine function first.

Sine Function

Sin (the sine function) takes an angle θ in a right-angled triangle and produces a ratio of the side opposite the angle θ to the hypotenuse.

Sin θ = Opposite / Hypotenuse

Inverse Sine

Inverse Sine Function

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

 

Example: 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°

Inverse Sine Formula

Let us consider 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:

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

Therefore, 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 functions and the question will be asked in the following way:

Problem: 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 angle “α ”

Step 1: Find the sin α°

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

Step 2: Now, for which angle sin α°  = 0.6293

Let’s find it out with Inverse sin:

α° = Sin-1 / (0.6293)

α°  = 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 Sine 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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