Inverse Sine Function

The inverse sine function is one of the inverse trigonometric functions which determines the inverse of the sine function and is denoted as Sin-1 or Arcsine.  For example: If the value of sine 90 degree is 1, then the value of inverse sin 1 or sin-1(1) will be equal to 90°. Each trigonometric function such as cosine, tangent, cosecant, cotangent and tangents has its inverse in a restricted domain. The inverse function formulas are used to calculate the measurement of angles with the help of the trigonometric ratios from the right-angle triangle. Generally, the sine inverse function is represented as sin-1. It does not state that sine cannot be raised to the negative power.


You can easily derive the  sine inverse function formula. Students can calculate the inverse of trigonometric function through the calculators also. You can easily find out the inverse of the function if you are aware of the six basic trigonometric functions. Through this article, you will study what is inverse sine, inverse sine function, inverse sine graph, inverse sine derivation, inverse sine examples, inverse sine formulas etc.


Inverse Sine Function

To understand the concept of inverse sine function clearly, we should first know about sine function.


Sine Function - The sine function of angle ϴ in the right-angle triangle is defined as the ratio of the opposite side of angle ϴ to the hypotenuse side.


Sin ϴ = \[\frac{\text{Opposite Side}}{\text{Hypotenuse side}}\]


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The inverse of sin function or Sin-1 also known as arcsin or asine obtains the angle ϴ if it takes the ratio \[\frac{\text{Opposite Side}}{\text{Hypotenuse side}}\].


Sine Inverse is represented by  Sin-1 or arcsin.


Sin Inverse Example

In a triangle PQR, PQ = 4.9 cm, QR = 4.0 cm and PR = 2.8 CM and angle Q = 35°


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Solution:


Sin 35° = \[\frac{\text{Opposite Side}}{\text{Hypotenuse side}}\]


Sin 35° = \[\frac{2.8}{4.9}\]


Sin 35° = 0.57


So, sin-1 \[\frac{\text{Opposite Side}}{\text{Hypotenuse side}}\] = 35°


Sin-1 (0.57) = 35°


Inverse Sine Formula

Let us understand the inverse sine formula through an example:

Let us consider that you want to calculate the depth of the deep sea-bed from the bottom of the sea and the following two specifications are given:

  1. The angle through which the cable makes with the deep sea–bed.

  2. The length of the cable

The sine function helps you to calculate the distance or depth of the ship from the deep sea-bed through the following method.


If the cable’s length is given as 40 m and the angle is 39°, then


Sin 39° = \[\frac{\text{Opposite Side}}{\text{Hypotenuse side}}\]


Sin 39° = \[\frac{d}{40}\]


Depth (d) = Sin 39° x 40


Depth (d) = 0.6293 x 40


Depth (d) = 25.172


Hence, the depth of the sea-bed is 25.17 cm.


Trigonometric Sine Function and Sine Inverse Formula Summary

The formula for trigonometric sine function for angle ϴ is stated as


Sin ϴ = \[\frac{\text{Opposite Side}}{\text{Hypotenuse side}}\]


The inverse sine formula is stated as


Sin-1 \[\frac{\text{Opposite Side}}{\text{Hypotenuse side}}\] = ϴ


Inverse Sine Graph


Arcsine function also known as the inverse of the sine function is represented as Sin-1 x. It is represented in the graph as shown below:


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Inverse Sine Derivative

Let us represent the function by ϴ = sin-1 k. We will make use of implicit differentiation to compute the derivative of this function.


Eliminate the inverse from the function


ϴ = sin-1 k → sin ϴ = k


The new function will be written in the form of:


k = sin ϴ


Where the domain of ϴ is restricted to the range of the principal values that the sin-1 function can select.


Differentiate the equation on both the sided in terms of ϴ


\[\frac{dk}{d\theta}\] = \[\frac{d}{\theta}\] (sin ϴ)


\[\frac{dk}{d\theta}\] = cos ϴ


Through the trigonometric identity, sin2 ϴ + cos2ϴ = 1 we get,


\[\frac{dk}{d\theta}\] = \[\sqrt{1 - sin^{2}\theta}\]


Using the inverse function, we can write the equations as,


\[\frac{dk}{d\theta}\] = \[\sqrt{1 - k^{2}}\]


Through the reciprocals of derivatives, we get


\[\frac{d\theta}{dk}\] = \[\frac{1}{\frac{dk}{d\theta}}\]


Here, you can see  the values of inverse sine in tabulated form.


Inverse Sine Value Table


ϴ

Sin-1 ϴ Values

Sin-1 ϴ (In Degrees)

-1

- \[\frac{\pi}{2}\]

-90°

-\[\frac{\sqrt{3}}{2}\]

- \[\frac{\pi}{3}\]

-60°

-\[\frac{\sqrt{2}}{2}\]

- \[\frac{\pi}{4}\]

-45°

-\[\frac{1}{2}\]

- \[\frac{\pi}{6}\]

-30°

0

0

\[\frac{1}{2}\]

\[\frac{\pi}{2}\]

30°

\[\frac{\sqrt{2}}{2}\]

\[\frac{\pi}{3}\]

45°

\[\frac{\sqrt{3}}{2}\]

\[\frac{\pi}{4}\]

60°

1

\[\frac{\pi}{6}\]

90°

 

Solved Example

1. Calculate the value of sin-1 ( sin (\[\frac{\pi}{6}\]))


Solution: Using the identity sin-1 (Sin(ϴ)= x), we get


Sin-1(sin (\[\frac{\pi}{6}\]) )= \[\frac{\pi}{6}\].


2. Find the value of 6 sin-11


Solution: Let X = sin-1, then sin X = 1


As we know, sin \[\frac{\pi}{2}\] = 1


6 sin-11 = 6 × \[\frac{\pi}{2}\]


6 sin-11 = 3 π


3. Find the value of sin ( cos-1 3/5)


Solution:


Suppose that cos-1 \[\frac{3}{5}\] = ϴ


So, cos ϴ = \[\frac{3}{5}\]


As we know that, sin ϴ = \[\sqrt{1- cos^{2}\theta}\]


So, sin ϴ = \[\sqrt{1 - \frac{9}{25}} = \frac{4}{5}\]


It means that sin ϴ = sin (cos-1 \[\frac{3}{5}\]) = \[\frac{4}{5}\].


Quiz Time

1. What is the value of sin-1 (1/2)?

  1. \[\frac{\pi}{3}\]

  2. \[\frac{\pi}{6}\]

  3. – \[\frac{\pi}{3}\]

  4. – \[\frac{\pi}{6}\]


2. Is Sin-1(sin – \[\frac{\pi}{2}\])} is equivalent to 2 \[\frac{\pi}{3}\]

  1. True

  2. False


3. The principal value of Sin-1{ sin 5 \[\frac{\pi}{6}\]} is

  1. 5 \[\frac{\pi}{6}\]

  2. \[\frac{\pi}{6}\]

  3. 7 \[\frac{\pi}{6}\]

  4. None of these

FAQ (Frequently Asked Questions)

1. What is Known as Inverse Trigonometric Functions?

Inverse trigonometric functions are nothing but the inverse function of some basic trigonometric functions such as sine, cosine, tangent, cotangent, secant and cosecant. These are also known as arcus function, ant trigonometric function or cyclometric function. The inverse trigonometric function in trigonometry is used to calculate the angles through any of the trigonometric ratios. For a given value of the trigonometric function, the arc function produces the length of arc required to obtain that particular value. Generally, inverse trigonometric functions perform the opposite operation of trigonometric functions. The trigonometric functions are usually applied in the right-angled triangle. The six important trigonometric functions are used to find the measurement of an angle in a right-angle triangle when the measurements of two sides of a triangle are known.


Inverse trigonometric functions are widely used in the field of engineering, physics, navigation and geometry.


Types of Inverse Trigonometric functions

There are six types of inverse trigonometric functions. These are as follows:

Arcsine

Arccosine

Arctangent

Arccontangent

Arcsecant

Arccosecant

2. What are the Applications of Inverse Trigonometric Functions?

Inverse trigonometric functions are widely used in construction sites, engineering, and architecture. For example, an archeologist observed an ancient monument on the peak of a steep mountain. Although, they have heavy machinery with them but cannot bring the machines in a steep slope. They can calculate the elevation of the path with a view to estimate the best possible route for machinery to lift. If the route ends 300 ft above their present position, 500 ft horizontal and machinery can’t go at a greater angle than 50 degrees. In such a case, they can use an inverse trigonometric function to find the angle of the route. Based on their findings, they can find the best possible route to take up the mountain.