# 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 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°.

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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 Learn more on the topics: Important questions for Class 11 Maths Set Theory Symbols Area of a Triangle Value of Pi

#### 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?