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.
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 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°.
- 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 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)|
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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