Inverse sine is one of the inverse trigonometric functions and it is written as sin-1x and is read as "sin inverse of x". It is also known as arcsin (x). We have 6 inverse trigonometric functions such as
- arcsin x = sin-1x = inverse of sin x
- arccos x = cos-1x = inverse of cos x
- arctan x = tan-1x = inverse of tan x
- arccsc x = csc-1x = inverse of csc x
- arcsec x = sec-1x = inverse of sec x
- arccot x = cot-1x = inverse of cot x
Here, we will study in detail about the inverse sine function (arcsin) along with its graph, domain, range, and properties. Also we will learn the formulas, derivative, and integral of sin inverse x along with a few solved examples.
|1.||What is Inverse Sine?|
|2.||Domain, Range, and Graph of Inverse Sine|
|3.||Steps to Find Sin Inverse x|
|4.||Properties of Inverse Sine|
|5.||Derivative of Inverse Sine|
|6.||Integral of Inverse Sine|
|7.||FAQs on Inverse Sine|
What is Inverse Sine?
The inverse sine function is the inverse of the sine function and thus it is one of the inverse trigonometric functions. It is also known as arcsin function which is pronounced as "arc sin". It is mathematically written as "asin x" (or) "sin-1x" or "arcsin x". We read "sin-1x" as "sin inverse of x". We know that if two functions f and f-1 are inverses of each other, then f(x) = y ⇒ x = f-1(y). So sin x = y ⇒ x = sin-1(y). i.e., when "sin" moves from one side to the other side of the equation, it becomes sin-1. Let us consider a few examples to see how the inverse sine function works.
Inverse Sine Examples
- sin 0 = 0 ⇒ 0 = sin-1(0)
- sin π/2 = 1 ⇒ π/2 = sin-1(1)
- sin π/6 = 0.5 ⇒ π/6 = sin-1(0.5)
Inverse Sine Formula
In a right-angled triangle, the sine of an angle (θ) is the ratio of its opposite side to the hypotenuse. i.e., sin θ = (opposite side) / (hypotenuse). Then by the definition of inverse sine, θ = sin-1[ (opposite side) / (hypotenuse) ] .
Thus, the inverse sine function is used to find the angle in a right-angled triangle when the opposite side and the hypotenuse are given. Also, it can be used to find the unknown angles in any triangle by using the sine law. In a triangle ABC, if AB = c, BC = a, and CA = b, then by the law of sines,
(sin A) / a = (sin B) / b
sin A = (a sin B) / b
A = sin-1 [ (a sin B) / b ]
Likewise, we can find the other angles of the triangle.
Domain, Range, and Graph of Inverse Sine
In this section, let us see how can we find the domain and range of the inverse sine function. Also, we will see the process of graphing it.
Domain and Range of Inverse Sine
We know that the sine function is a function from R → [-1, 1]. But sine function is NOT one-one on the domain R and hence its inverse does not exist. For the sine function to be one-one, its domain can be restricted to one of the intervals [-3π/2, -π/2], [-π/2, π/2], [π/2, 3π/2], etc. Corresponding to each of these intervals, we get a branch of the inverse sine. But the domain of sine function is usually restricted to [-π/2, π/2] to make it one-one.
We know that the domain and range of a function are interchanged to be the range and domain of its inverse function respectively. Hence,
- the domain of sin inverse x is [-1, 1]
- the range of sin inverse x is [-π/2, π/2].
i.e., arcsin x (or) sin-1x : [-1, 1] → [-π/2, π/2]
Graph of Inverse Sine
The graph of the inverse sine function with its range to be principal branch [-π/2, π/2] can be drawn using the following table. Here, we have chosen random values for x in the domain of sin inverse x which is [-1, 1].
|x||y = sin-1x|
|-1||sin-1(-1) = -sin-1(1) = -π/2|
|-0.5||sin-1(-0.5) = -sin-1(0.5) = -π/6|
|0||sin-1(0) = 0|
|0.5||sin-1(0.5) = π/6|
|1||sin-1(1) = π/2|
By plotting these points on the graph, we get arcsin graph.
Steps to Find Sin Inverse x
Here are the steps to find the sin inverse of x.
- Since the range of sin inverse x is [-π/2, π/2], the answer should lie in this interval.
- Assume that y = sin-1x. Then by the definition of inverse sin, sin y = x.
- Think what value of y in the interval [-π/2, π/2] satisfies the equation sin y = x and that is the answer.
Here are some examples to understand these steps.
Examples of Finding Sin Inverse of x
Note that sin-1x should always result in some angle that lies in the interval [-π/2, π/2].
- sin-1(1) = π/2 as sin π/2 = 1
- sin-1(-1) = -π/2 as sin (-π/2) = -1
- sin-1(-0.5) = -π/6 as sin (-π/6) = -0.5
Properties of Inverse Sine
Here are some properties/formulas of inverse sin. These are very helpful in solving the problems related to inverse sin in trigonometry.
- sin(sin-1x) = x only when x ∈ [-1, 1]
([When x ∉ [-1, 1], sin(sin-1x) is NOT defined)
- sin-1(sin x) = x, only when x ∈ [-π/2, π/2]. To know how to calculate sin-1(sin x) when x ∉ [-π/2, π/2], click here.
- sin-1(-x) = -sin-1x, when x ∈ [-1, 1]
- sin-1(1/x) = csc-1x, when |x| ≥ 1
- sin-1x + cos-1x = π/2, when x ∈ [-1, 1]
- sin-1(2x √1 - x²) = 2 sin-1x, when -1/√2 ≤ x ≤ 1/√2 and
sin-1(2x √1 - x²) = 2 cos-1x, when 1/√2 ≤ x ≤ 1
Derivative of Inverse Sine
cos y (dy/dx) = 1
dy/dx = 1/cos y ... (1)
Now, we have sin2y + cos2y = 1 ⇒ cos2y = 1 - sin2y ⇒ cos y = √1 - sin²y = √1 - x²
Substituting this in (1),
dy/dx = 1/√1 - x²
Thus, the inverse sine derivative (or) the derivative of sin inverse x is 1/√1 - x².
Integral of Inverse Sine
We will find ∫ sin-1x dx using the integration by parts. For this, we write the above integral as
∫ sin-1x · 1 dx
Using LIATE, f(x) = sin-1x and g(x) = 1.
By integration by parts,
∫f(x) . g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) ∫g(x) dx) dx + C
∫ sin-1x · 1 dx = sin-1x ∫1 dx - ∫ [d/dx(sin-1x) ∫x dx] + C
∫ sin-1x dx = sin-1x (x) - ∫ [1/√1 - x²] x dx + C
Let us evaluate the integral on the right side using u-substitution method. For this, assume that 1 - x2 = u. From this, -2x dx = du (or) x dx = -1/2 du.
∫ sin-1x dx = x sin-1x - ∫(1/√u) (-1/2) du + C
= x sin-1x + 1/2 ∫u-1/2 du + C
= x sin-1x + (1/2) (u1/2/(1/2)) + C
= x sin-1x + √u + C
= x sin-1x + √1 - x² + C
Therefore, ∫ sin-1x dx = x sin-1x + √1 - x² + C.
Important Notes on Inverse Sine:
- Inverse sine can be written as sin-1 (or) arcsin (or) asin and it is a function with domain [-1, 1] and range [-π/2, π/2].
- Inverse sin is NOT same as (sin x)-1 as (sin x)-1 = 1/(sin x) = csc x.
- sin(sin-1x) is NOT always x. sin(sin-1x) = x only when x ∈ [-1, 1].
- sin-1(sin x) is NOT always x. sin-1(sin x) = x only when x ∈ [-π/2, π/2].
Related Topics to Sin Inverse x:
Examples on Sin Inverse x
Example 1: If θ is an acute angle in a right triangle whose opposite side is 4 units and the hypotenuse is 5 units, find θ.
We know that sin θ = (opposite side) / (hypotenuse) = 4/5.
From the definition of inverse sine,
θ = sin-1 (4/5) ≈ 53.13.
(This is by using the calculator.)
Answer: θ ≈ 53.13.
Example 2: Find the values of the following: a) sin (sin-1 2) b) sin-1(sin 7π/6).
a) We know that sin(sin-1 x) is NOT defined when x is NOT in [-1, 1] as the domain of inverse sine is [-1, 1].
Thus, sin(sin-1 2) is NOT defined.
b) sin-1(sin 7π/6)
Note that this is NOT equal to 7π/6 as 7π/6 ∉ [-π/2, π/2].
So we have to convert this angle to lie in the interval [-π/2, π/2].
We have sin 7π/6 = sin (π + π/6) = - sin π/6
Thus, sin-1(sin 7π/6) = sin-1(- sin (π/6))
We know that sin-1(-x) = - sin-1x. So
sin-1(sin 7π/6) = - sin-1(sin (π/6)) = -π/6
Answer: a) sin(sin-1 2) is NOT defined b) sin-1(sin 7π/6) = -π/6.
Example 3: Prove the following: 3 sin-1x = sin-1(3x - 4x3), for x ∈ [-1/2, 1/2].
Let us substitute x = sin θ in the RHS.
RHS = sin-1(3 sin θ - 4sin3θ)
From trigonometric formulas, we have
sin 3θ = 3 sin θ - sin3θ
So RHS = sin-1(sin 3θ)
= 3 sin-1x (because x = sin θ)
Answer: We have proved that 3 sin-1x = sin-1(3x - 4x3).
FAQs on Inverse Sine
What is Sin Inverse of x?
Sin inverse of x is the inverse of the sine function. i.e., if y = sin x then x = sin-1(y). Here, sin-1 is the inverse function of sin.
How to Find the Inverse Sine of x?
To find the inverse sine of any number, just see what angle of sine gives that number. For example, sin-1(1/√2) = π/4 as sin π/4 = 1/√2. But make sure that the angle lies in the interval [-π/2, π/2].
What is the Inverse Sine of 1?
We know that sin π/2 = 1. Then by the definition of inverse sine, sin-1(1) = π/2. i.e., the value of inverse sine of 1 is π/2.
Is the Inverse of Sine Csc?
No, the inverse of sine is not cosec. In fact, the inverse of sine is sin-1 (or) arcsin function. But note that (sin x)-1 = 1/(sin x) = csc x but this is not the inverse of the sine function.
How to Write Inverse Sine?
Inverse sine of x is written in one of the following ways:
- arcsin (x)
- asin (x)
Why do We Use the Inverse Sine?
The inverse sine function is used to find the angles in a right triangle when its opposite side and hypotenuse are known. i.e., angle = sin-1(opposite side to the angle/hypotenuse).
What is the Derivative of Sin Inverse x?
The derivative of sin-1x is 1/√1 - x². It is mathematically written as d/dx(sin-1x) = 1/√1 - x² (or) (sin-1x)' = 1/√1 - x².
What is the Integral of Inverse Sin?
The integral of sin-1x is x sin-1x + √1 - x² + C. It is mathematically written as ∫ sin-1x dx = x sin-1x + √1 - x² + C.
What is Inverse Sine of 2?
The domain of the inverse sine is [-1, 1]. Thus, it cannot take the value 2. So the inverse sine of 2 is not defined.