Arcsin

Arcsin is one of the six main inverse trigonometric functions. It is the inverse trigonometric function of the sine function. Arcsin is also called inverse sine and is mathematically written as arcsin x or sin^-1x (read as sine inverse x). An important thing to note is that sin^-1x is not the same as (sin x)^-1, that is, sin^-1x is not the reciprocal function of sin x. In inverse trigonometry, we have six inverse trigonometric functions - arccos, arcsin, arctan, arcsec, arccsc, and arccot.

Arcsin x gives the measure of the angle corresponding to the ratio of the perpendicular and hypotenuse of a right-angled triangle. In this article, we will explore the concept of arcsin and derive its formula. We will also discuss the domain and range of arcsin x and hence, plot its graph. We will also solve various examples using the identities of arcsin x to understand its applications and the concept better.

Arcsin is the inverse trigonometric function of the sine function. It gives the measure of the angle for the corresponding value of the sine function. We denote the arcsin function for the real number x as arcsin x (read as arcsine x) or sin^-1x (read as sine inverse x) which is the inverse of sin y. If sin y = x, then we can write it as y = arcsin x. Arcsin is one of the six important inverse trigonometric functions. The six inverse trigonometric functions are:

• Arcsin: Inverse of sine function, denoted by arcsin x or sin^-1x
• Arccos: Inverse of cosine function, denoted by arccos x or cos^-1x
• Arctan: Inverse of tangent function, denoted by arctan x or tan^-1x
• Arccot: Inverse of cotangent function, denoted by arccot x or cot^-1x
• Arcsec: Inverse of secant function, denoted by arcsec x or sec^-1x
• Arccsc: Inverse of cosecant function, denoted by arccsc x or csc^-1x

The arcsin function helps us find the measure of an angle corresponding to the sine function value. Let us see a few examples to understand its functioning. We know the values of the sine function for some specific angles using the trigonometric table.

• If sin 0 = 0, then arcsin 0 = 0
• sin π/6 = 1/2 implies arcsin (1/2) = π/6
• sin π/3 = √3/2 implies arcsin (√3/2) = π/3
• If sin π/2 = 1, then arcsin (1) = π/2

We can use the arcsin formula when the value of sine of an angle is given and we want to evaluate the exact measure of the angle. Consider a right-angled triangle. We know that sin θ = Opposite Side / Hypotenuse. As arcsin is the inverse function of the sine function, therefore, we have θ = arcsin (Opposite Side / Hypotenuse). Therefore, the formula for arcsin x is,

θ = arcsin (Opposite Side / Hypotenuse)

We can also use the law of sines to derive the arcsin formula. For a triangle ABC with sides AB = c, BC = a and AC = b, we have sin A / a = sin B / b = sin C / c. Then, taking two at a time, we have

sin A / a = sin B / b

⇒ sin A = (a/b) sin B

⇒ A = arcsin [(a/b) sin B]

Similarly, we can find the measure of the angles B and C using the same method.

Now that we know the arcsin formula, we will plot the graph of arcsin x using some of its points. As discussed the functioning of arcsin, we know the values of the sine function for some specific angles and using trigonometric formulas, we have

• sin 0 = 0 implies arcsin 0 = 0 → (0, 0)
• sin π/6 = 1/2 implies arcsin (1/2) = π/6 → (1/2, π/6)
• sin π/3 = √3/2 implies arcsin (√3/2) = π/3 → (√3/2, π/3)
• sin π/2 = 1 implies arcsin (1) = π/2 → (1, π/2)
• sin (-π/4) = -1/√2 implies arcsin (-1/√2) = -π/4 → (-1/√2, -π/4)
• sin (-π/6) = -1/2 implies arcsin (-1/2) = -π/6 → (-1/2, -π/6)

Now, by plotting the above points on a graph, we have the graph of arcsin given below:

As we know that two functions are inverses of each other if they are bijective and the domain and range of the function become the range and domain, respectively of the inverse function. We know that the domain of sin x is all real numbers and its range is [-1, 1]. But with this domain, sin x is not bijective. So, we restrict the domain of sine function to [–π/2, π/2], then sin x becomes bijective with domain [–π/2, π/2] and range [-1, 1]. When the domain of sin x is restricted to [–3π/2, –π/2], [–π/2, π/2], or [π/2, 3π/2], and so on, and range [-1, 1], then sin x is bijective and hence, correspondingly we can define arcsin with domain [-1, 1] and range [–3π/2, –π/2], [–π/2, π/2], or [π/2, 3π/2], and so on.

We get different branches of the arcsin function for each interval. The branch of arcsin corresponding to domain [-1, 1] and range [–π/2, π/2] is called the principal value branch. So, the arcsin is defined as arcsin: [-1, 1] → [–π/2, π/2]. Hence, the domain and range of arcsin are:

• Domain of Arcsin: [-1, 1]
• Range of Arcsin: [–π/2, π/2]

Now, we will discuss some of the important properties and identities of the arcsin function that help us to simplify and solve various problems in trigonometry.

• sin (arcsin x) = x, if x is in [-1, 1]
• arcsin (sin x) = x, if x is in [–π/2, π/2]
• arcsin (1/x) = arccsc x, if x ≤ -1 or x ≥ 1
• arcsin (–x) = - arcsin x, if x ∈ [-1, 1]
• arcsin x + arccos x = π/2, if x ∈ [-1, 1]
• 2 arcsin x = arcsin (2x √(1 - x²)), if -1/√2 ≤ x ≤ 1/√2
• 2 arccos x = arcsin (2x √(1 - x²)), if 1/√2 ≤ x ≤ 1
• arcsin x + arcsin y = arcsin [x√(1 - y²) + y√(1 - x²)]

Important Notes on Arcsin

• Arcsin is the inverse function of sine function.
• The domain and range of arcsin are [-1, 1] and [–π/2, π/2], respectively.
• The derivative of arcsin is 1/√(1 - x²).
• The integral of arcsin is ∫arcsin x dx = x sin^-1x + √(1 - x²) + C

☛ Related Topics:

• Sin 1 in Degrees
• Inverse Trigonometric Ratios
• Inverse Trig Derivatives

What is Arcsin in Trigonometry?

Arcsin is an inverse trigonometric function of the sine function. We denote the arcsin function for the real number x as arcsin x (read as arcsine x) or sin^-1x (read as sine inverse x). It is one of the six main inverse trigonometric functions give by, arccos, arcsin, arctan, arcsec, arccsc, and arccot. An important thing to keep in mind is that sin^-1x is not the reciprocal of sine.

What is Arcsin Formula?

The formula for arcsin is given by, θ = arcsin (Opposite Side / Hypotenuse), where θ is the angle in a right-angled triangle. The arcsin function helps us find the measure of an angle corresponding to the sine function value. We can also find the measure of an angle in a triangle using the arcsin formula derived using the law of sines.

What is the Derivative of Arcsin x?

The derivative of arcsin is given by, d/dx(arcsin x) = 1/√(1 - x²). We can derive this formula using the first principle of derivatives and the chain rule method of differentiation.

How to Integrate Arcsin?

The integral of arcsin is given by, ∫arcsin x dx = x sin^-1x + √(1 - x²) + C, where C is the constant of integration. It can be derived using different methods such as integration by parts and substitution method followed by integration by parts.

What is the Domain and Range of Arcsin?

The domain and range of arcsin are:

• Domain of Arcsin: [-1, 1]
• Range of Arcsin: [–π/2, π/2]

We restrict the domain of the sine function to [–π/2, π/2] to make it bijective and hence, define the arcsin function as two functions are inverses of each other if they are bijective. The branch of arcsin corresponding to domain [-1, 1] and range [–π/2, π/2] is called the principal value branch.

How to Plot the Arcsin Graph?

Using the definition and functioning of arcsin, we can plot some points on the graph with the help of a trigonometric table. Some of the points are:

• sin 0 = 0 implies arcsin 0 = 0 → (0, 0)
• sin π/6 = 1/2 implies arcsin (1/2) = π/6 → (1/2, π/6)
• sin π/3 = √3/2 implies arcsin (√3/2) = π/3 → (√3/2, π/3)
• sin π/2 = 1 implies arcsin (1) = π/2 → (1, π/2)
• sin (-π/4) = -1/√2 implies arcsin (-1/√2) = -π/4 → (-1/√2, -π/4)
• sin (-π/6) = -1/2 implies arcsin (-1/2) = -π/6 → (-1/2, -π/6)

Then, by plotting these points and joining through a curve, we get the arcsin graph.

Is Arcsin the Inverse of Sin?

Arcsin is the inverse of the trigonometric function sin. When the arcsin function is defined as arcsin: [-1, 1] → [–π/2, π/2], then we say that it is the inverse of sin: [–π/2, π/2] → [-1, 1].

What is the Difference between Sin and Arcsin?

Sine is a trigonometric function that maps a real number to an angle whereas arcsin is the inverse of the sine function. Both functions are defined as arcsin: [-1, 1] → [–π/2, π/2], then we say that it is the inverse of sin: [–π/2, π/2] → [-1, 1] and are inverses of each other.

Why Arcsin (-2) is Not Defined?

Arcsin (-2) is not defined because the domain of arcsin is restricted to [-1, 1] and -2 does not lie in the interval [-1, 1].

What are the Identities of Arcsin?

Some of the important formulas and identities of arcsin are:

• sin (arcsin x) = x, if x is in [-1, 1]
• arcsin (sin x) = x, if x is in [–π/2, π/2]
• arcsin (1/x) = arccsc x, if x ≤ -1 or x ≥ 1
• arcsin (–x) = - arcsin x, if x ∈ [-1, 1]
• arcsin x + arccos x = π/2, if x ∈ [-1, 1]
• 2 arcsin x = arcsin (2x √(1 - x²)), if -1/√2 ≤ x ≤ 1/√2

What is Arcsin of Sin?

The formula for arcsin of sin is given by, arcsin (sin x) = x, if x is in [–π/2, π/2].