Inverse Sine

The inverse Sine function is a trigonometric function which expresses the inverse of the sine function and is represented as  Sin-1 or Arcsine. If sin 90 degrees is equal to 1, then the inverse of sin 1 or sin-1 (1) equals to 90 degrees. Every trigonometric function, whether it is Sine, Cosine, Tangent, Cotangent, Secant or Cosecant has an inverse of it, though in a restricted domain. The function of inverse or inverse function is used to determine the angle measure, with the use of basic trigonometric ratios from the right triangle. Mostly, the inverse sine function is represented using sin-1. It does not mean that sine is not raised to the negative power.

The formula for inverse sine function is simple to derive. Students can use the calculators also, which are available online, to find the inverse of trig functions. If you are thorough with the six basic trigonometric functions, the inverse of the functions can be easily found. Let us learn in detail the concept of  Sin-1 here.

Inverse Sine Function

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: 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°.

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.

Summary of Formula:

The formula for the trigonometric sine function is given by:

sin (θ) Opposite Side/ Hypotenuse

The inverse sine function formula or the arcsin formula is given as:

sin-1 (Opposite side/ hypotenuse) = θ

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

The derivative of inverse sine function is given by: d/dx Sin-1x= 1 / √(1-x2)

Let us proof this equation.

Proof: f(x) =sin(x) and g(x) = sin-1x

If we differentiate g(x) with respect to x, we get;

g'(x) = 1/f'(g(x)) = 1/cos(sin-1x) …………….(1)

Now we know, the inverse of sine function,  y = Sin-1x

or sin y = x  …………(2)

Hence we can write, the derivative as;

g'(x) = 1/cos y   (from eq.1)

Since, we know, by trigonometric identities;

cos2 y + sin2 y = 1

Therefore, we can write;

cos y = √(1-sin2 y) =  √(1 – x2)          (since sin y = x)

Now, again putting this value in the derivative;

g'(x) =  1/cos y = 1/√(1 – x2)

Thus,

d/dx(sin-1x) = 1/√(1 – x2)

Hence, proved.

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 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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