Inverse Sine

Inverse Sine function is one of the important inverse trigonometric functions. The inverse of trigonometric functions gives the angle with the help of the sides of the triangles. In this article, we will explore the inverse sine function, its definition, formula, and properties. We will also discuss the domain and range of the inverse sine function, the graph of the inverse sine function, the derivative of inverse sine, and the integral of inverse sine along with the solved examples. Let’s start our learning on the topic of “Inverse Sine”.

What is Sine Function?

The sine function in a right-angle triangle shows the relationship between the length of the side opposite an angle (let’s call it θ) and the length of the hypotenuse. It’s expressed as Sin θ = Opposite side/Hypotenuse. This formula helps us understand the sine function in a triangle.

What is Inverse Sine Function?

The inverse sine function is like a math operation that does the opposite of the regular sine function. It’s part of a group of functions called inverse trigonometric functions. Another name for it is arcsine which is pronounced as “arc sin”. In mathematical terms, it can be written as “asin x”, “sin^-1x”, or “arcsine x”. When written as “sin-1x,” it means “sin inverse of x.”

In an equation like sin x = y, one can use the inverse sine function to find x. So, if sin x = y, then x = sin^-1(y). It’s like switching sin from one side of the equation to the other.

Inverse Sine Definition

The inverse sine function is a mathematical operation that, when applied to a certain value, gives another value such that if you take the sine of that result, you get back to the original value. It is like a mathematical operation that undoes the sine operation, allowing you to find the angle associated with a given sine value.

y = sin x

⇒ x = sin^-1y

Inverse Sine Formula

In a right-angled triangle, the sine of an angle (θ) is the ratio of the opposite side to the hypotenuse (sin θ = opposite side / hypotenuse). The inverse sine function, denoted as θ = sin⁻¹[(opposite side) / (hypotenuse)], helps find the angle when the opposite side and hypotenuse are known. This function is expressed as θ = arcsin[(opposite side) / (hypotenuse)]. It’s used not only in right-angled triangles but also in any triangle through the sine law. In a triangle ABC, where AB = c, BC = a, and CA = b, sin A = (a sin B) / b, and A = sin⁻¹[(a sin B) / b].

θ = sin^-1 (opposite side/hypotenuse)

This formula is used in the context of a right-angled triangle, where θ represents an angle, and the inverse sine function helps find the angle when the lengths of the opposite side and the hypotenuse are known. The result θ is the measure of the angle in degrees or radians, depending on the context.

Right angled triangle

Read more about Inverse Trigonometric Functions

Properties of Inverse Sine Function

Inverse sine function has the following main properties:

1. sin (sin^-1 x) = x, valid for −1 ≤ x ≤ 1.

2. sin^-1(sin x) = x, valid for -π/2 ≤ x ≤ π/2

3. sin⁻¹(−x) =−sin⁻¹ x, valid for −1 ≤ x ≤1.

4. sin^-1(1/x) = csc^-1 x, valid for |x| ≥ 1

5. sin^-1x + cos^-1x = π/2, valid for (-1 ≤ x ≤ 1)

Domain and Range of Inverse Sine

The domain is the input value for which inverse sine function is defined and range is the corresponding value for that domain. The domain and range of Inverse sine function is discussed below:

Domain of Inverse Sine

The domain of the inverse sine function, often denoted as sin^-1x, is the set of real numbers from -1 to 1, inclusive. In mathematical terms, it is represented as [-1, 1]. This means that you can input any value within this range into the inverse sine function.

Range of Inverse Sine

The range of the inverse sine function, denoted as sin{-1}x or arcsin x, is the set of real numbers from -π/2 to π/2, inclusive. In mathematical terms, it is represented as -π/2, π/2. This signifies the possible output values or angles that the inverse sine function can yield when given inputs from the domain [-1, 1].

How to Find Sin Inverse x

To find the sin inverse of x:

1. Recognize that the possible answers lie in the range of [-π/2, π/2].
2. Let’s assume y is the sin inverse of x, denoted as y = sin⁻¹x. According to the definition of inverse sine, this means sin y = x.
3. Consider values of y within the interval [-π/2, π/2] that satisfy the equation sin y = x. The solution lies in this range.

Graph of Inverse Sine Function

The graph of the inverse sine function, often denoted as y = sin^-1(x) or y = arcsin(x), is a curve that represents the values of the inverse sine for different inputs. The domain of the inverse sine function is [-1, 1], and its range is (-π/2, π/2) or (-90°, 90°) in degrees.

Here is a table of values and the corresponding graph:

x	Sin-1x (in radians)	Sin-1x (in degrees)
-1	– π/2	90
-0.5	– π/6	30
0	0	0
0.5	π/6	30
1	π/2	90

The graph starts at (-1, -π/2) and (1,π/2), representing the minimum and maximum values for sin-1x.

The curve is symmetric with respect to the y-axis, reflecting the symmetry of the sine function.

As (x) approaches -1 or 1, sin-1(x) approaches -π/2 or π/2, respectively.

The graph is confined to the interval [-1, 1] on the x-axis.

When graphed, the inverse sine function produces a curve that starts at -1, -π/2, increases to the peak at (0, 0), and then descends to (1, π/2). This curve represents the angles whose sine values correspond to the given x-values in the table.

Derivative and Integral of Inverse Sine

In this section we will discuss about the derivative and integral of the inverse sine function.

Derivative of Inverse Sine

The derivative of the inverse sine function, denoted as (y = sin^-1x), is found by considering the relationship (sin y = x). Using the chain rule, the derivative is expressed as:

dy/dx = 1/cos y

Further, by using the Pythagorean identity (sin²y + cos²y = 1) the expression for cos y is substituted:

(dy / dx) = 1 / √(1 – x²)

Therefore, the derivative of the inverse sine function, or sin^-1x, is 1 / √ (1 – x²)

Integral of Inverse Sine

Find the integral of sin^-1x with respect to (x) using the integration by parts method. To start, we express the integral as:

∫ sin^-1x. 1 dx

Following the LIATE rule, where f(x) = sin^-1x and g(x) = 1, the integration by parts formula is applied:

∫ f(x). g(x) dx = f(x) ∫ g(x)dx – ∫ [{(d/dx) f(x)}. ∫ g(x) dx] dx + C

Substituting f(x) = sin^-1x and g(x) = 1, we get:

∫ f(x). g(x) dx = f(x) ∫ g(x)dx – ∫ [{(d/dx) f(x)}. ∫ g(x) dx] dx + C

Now, we evaluate the inner integral using the u-substitution method. Let (u = 1 – x²), then (-2xdx = du) (or x dx = -1/2 du). Substituting these, we get:

∫sin^-1 dx = xsin^-1x + ∫ (1 / √u)(-1/2) du}+ C

∫sin^-1 dx = xsin^-1x – √u+ C

Replace (u) with 1-x², we get:

∫sin^-1 dx = xsin^-1x + √(1 – x²) + C

Therefore, the integral of sin^-1x with respect to (x) is xsin^-1x + √(1 – x²) + C

Solved Examples on Inverse Sine

Example 1: A ladder is leaning against a wall. If the angle between the ladder and the ground is π/6 radians, and the ladder is 5 meters long, find the height the ladder reaches on the wall.

Solution:

Let (h) be the height the ladder reaches on the wall. We can use the inverse sine function to relate the angle π/6 to the ratio of the opposite side (height) to the hypotenuse (ladder length):

Sin^-1 (h/5)= π/6

Solving for (h)

h= 5 sin π/2

h= 5/2

Therefore, the height the ladder reaches on the wall is 5/2 meters.

Example 2: Sinθ= 3/5 and (θ) is in the first quadrant, find Sin^-1 3/5

Solution:

The problem provides the sine value of an angle, and we need to find the angle itself using the inverse sine function.

In this case,

Sin θ= 3/5.

We can write this as:

θ = Sin^-1 (3/5)

By calculating, θ = Sin^-1 (3/5) is approximately 0.6435 radians or 38.68° (rounded to two decimal places).

Therefore,

Sin^-1 (3/5) is 0.6435 radians or 38.68° in the first quadrant.

Inverse Sine – Practice Questions

1. Evaluate sin^-1 (½) .

2. Find the exact value of (x) in sin^-1x= π/4.

3. Calculate cos (sin^-1 ⅗)

4. Solve for (x) in the equation sin^-1(2x) = π/3

5. Determine the domain and range of the function f(x) = sin^-1x (3x – 1).

Inverse Sine – FAQs

1. What is Inverse Sine?

Inverse sine, denoted as Sin^-1 or arcsin, is a mathematical function that gives the angle whose sine is a specified value. It’s the opposite operation of the sine function.

2. What is the Value of Inverse Sine of 1?

The value of Sin^-11 is π/2 radians or 90°. This is because sin of π/2 is equal to 1.

3. What is the difference between Sin^-1 and 1/Sin x?

Sin^-1x represents the inverse sine function, providing an angle whose sine is x. On the other hand, 1/sinx represents the cosecant function (cosec x), which is the reciprocal of the sine function.

4. Is Sine Inverse equal to Cosec Function?

No, Sin^-1x and cosec x is not equal. Sin^-1x gives an angle whose sine is x, while cosec x is the reciprocal of the sine function, 1/sinx

5. What is the Integral of Inverse Sin?

The integral of Sin^-1x with respect to x is xSin^-1 x + (√1-x²)+ C, where C is the constant of integration.

6. Why do We Use the Inverse Sine Function?

The inverse sine function helps find angles in a right-angled triangle when the lengths of the sides are known. It is also useful in solving trigonometric equations and understanding the relationship between angles and sine values.

7. What is the Value of Inverse Sine of 0?

The value of inverse sine of 0 is 0°.