Inverse Sine

Inverse Sine is a trigonometric function which denotes the inverse of the sine function and is represented as  Sin-1 or Arcsine. The formula for this function is simple to derive. Students can use the calculators also, which are available online, to find the inverse of trig functions. 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 function 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

The inverse of sine function or Sin-1   takes the ratio, Opposite Side / Hypotenuse Side and produces angle θ. It is also written as arcsin or asine.

Sin inverse is denoted by sin-1 or arcsin.

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.

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 Sine 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)

Inverse Sine Table

θ Sin-1 or Arcsin(θ) (in Radian) Sin-1 or Arcsin(θ) (in Degree)
-1 -π/2 -90°
-√3/2 -π/3 -60°
-√2/2 -π/4 -45°
-1/2 -π/6 -30°
0 0
1/2 π/6 30°
√2/2 π/4 45°
√3/2 π/3 60°
1 π/2 90°

Example

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?

Solution: 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 “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°

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