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. 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.
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
- 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°.
- 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 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°
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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