What is Inverse Sine Function?

The inverse sine function, denoted as “arcsin” or “sin^(-1)”, is a mathematical function that undoes the operation of the sine function. In other words, it allows you to find the angle (in radians or degrees) that, when passed through the sine function, produces a given value. The arcsin function is the inverse of the sine function and is defined for values between -1 and 1.

Mathematically, if you have a value “x” in the range -1 to 1, and you want to find the angle “θ” that satisfies the equation:

sin(θ) = x

You can use the arcsin function to find θ:

θ = arcsin(x)

The result θ is typically expressed in radians or degrees, depending on the context. The arcsin function returns values in the range from -π/2 to π/2 radians (-90 degrees to 90 degrees), as the sine function is periodic with a period of 2π.

For example, if you want to find the angle θ such that sin(θ) = 0.5, you can use the arcsin function:

θ = arcsin(0.5) ≈ 30 degrees or π/6 radians

So, θ is approximately 30 degrees or π/6 radians.

The arcsin function is one of the inverse trigonometric functions, which also include arccos (inverse cosine), arctan (inverse tangent), and others. These functions are useful in trigonometry and various mathematical and engineering applications to solve problems involving angles and triangles.

What is Sine Function?

The sine function, often denoted as “sin,” is a fundamental trigonometric function in mathematics. It relates the angle of a right triangle to the ratio of the length of the side opposite that angle to the length of the hypotenuse (the longest side of the triangle).

In a right triangle, if you have an angle θ, the sine of that angle is defined as:

sin(θ) = (opposite side) / (hypotenuse)

Here’s a brief explanation of the components:

θ: The angle in the right triangle for which you want to find the sine.
The “opposite side” is the side of the triangle that is opposite to the angle θ.
The “hypotenuse” is the longest side of the right triangle and is opposite the right angle.

The sine function generates a periodic wave that oscillates between -1 and 1 as the angle θ varies. It’s a fundamental tool in trigonometry and is widely used in various fields of science and engineering to model periodic phenomena, such as sound waves, light waves, and oscillatory motion.

The sine function has a specific shape when graphed, producing a smooth, continuous wave that repeats itself every 360 degrees (or 2π radians) due to its periodic nature. The sine function is crucial in trigonometric identities, calculus, and various mathematical and scientific applications.

READ Principle of Mathematical Induction

Inverse Sine Formula

The inverse sine function, often denoted as “sin^(-1)” or “arcsin,” is the inverse of the sine function. It is used to find the angle whose sine value is known. The inverse sine function takes an input between -1 and 1 and returns an angle in the range of -π/2 to π/2 radians (-90 degrees to 90 degrees).

The formula for the inverse sine function is:

sin^(-1)(x) = y

Where:

x is the sine value of an angle, which should be in the range -1 ≤ x ≤ 1.
y is the angle in radians that satisfies sin(y) = x, and it will be in the range -π/2 ≤ y ≤ π/2 radians (-90 degrees ≤ y ≤ 90 degrees).

Here are some key points to keep in mind when using the inverse sine function:

The input ‘x’ must be within the range -1 ≤ x ≤ 1. If ‘x’ is outside this range, there is no real solution for the inverse sine, and you’ll get an error.
The output ‘y’ will be in radians. If you want the result in degrees, you can convert it using the relation: 1 radian = 180/π degrees.
The inverse sine function has multiple solutions for a given ‘x’ value. In general, it has two solutions: one in the first quadrant (0 ≤ y ≤ π/2) and one in the fourth quadrant (-π/2 ≤ y ≤ 0). The specific value returned depends on the convention used, but it is often the one in the first quadrant.

Here’s an example of using the inverse sine function:

If sin(y) = 0.5, to find the angle ‘y’ in radians: y = sin^(-1)(0.5) ≈ 0.5236 radians (or approximately 30 degrees).

Keep in mind that trigonometric functions and their inverses are widely used in various fields, including geometry, physics, and engineering, to solve problems involving angles and lengths in triangles and periodic phenomena.

Domain and Range of Inverse Sine

The inverse sine function, often denoted as “arcsin” or “sin^(-1),” is the inverse of the sine function. The domain and range of the inverse sine function are as follows:

Domain:
- The domain of the inverse sine function is the set of real numbers, which means you can input any real number into the inverse sine function. In mathematical notation, the domain is often expressed as: Domain of arcsin(x): x ∈ [-1, 1]
The reason for the domain being limited to the interval [-1, 1] is because the sine function’s values are bounded within this range. The sine function produces values between -1 and 1 for any real input, and the inverse sine function “undoes” the sine function’s operation.
Range:
- The range of the inverse sine function is typically the open interval from -π/2 to π/2, which is (-π/2, π/2). In mathematical notation: Range of arcsin(x): y ∈ (-π/2, π/2)
This means that the output of the inverse sine function will be an angle in radians that falls within this range. In degrees, this corresponds to the open interval from -90° to 90°.

Keep in mind that the specific notation and conventions for the domain and range may vary in different contexts or textbooks, but the intervals mentioned here are commonly used to describe the domain and range of the inverse sine function.

READ If you place a 29-foot ladder against the top of a building and the bottom of the ladder is 24 feet from the bottom of the building, How tall is the building? Round to the nearest tenth of a foot.

Steps to Find Sin Inverse x

To find the inverse sine (sin^(-1) or arcsin) of a value “x,” follow these steps:

Ensure that the value of “x” is within the range of the sine function: -1 ≤ x ≤ 1. Sine only takes values within this range, and attempting to find the arcsine of a value outside this range is undefined.
Identify the value “x” for which you want to find the arcsine.
Use the arcsin function on your calculator or a mathematical software program to find the inverse sine value. The symbol for the arcsin function is sin^(-1) or asin. You can usually find it on your calculator by pressing the “2nd” or “shift” key followed by the sine function key, or by using the “sin^(-1)” button if available.
Enter the value of “x” into the arcsin function and press the appropriate key to calculate the result.
The result of the arcsin function will be an angle in radians between -π/2 and π/2 (or between -90 degrees and 90 degrees) that corresponds to the given value “x.” It represents the angle whose sine is equal to “x.”

Keep in mind that the result may be expressed in radians, so if you want the answer in degrees, you can convert it using the following relationship:

Radians to Degrees: Degrees = (Radians × 180) / π

This will give you the angle in degrees that corresponds to the given value “x” when taking the inverse sine.

Derivative of Inverse Sine

The derivative of the inverse sine function, denoted as arcsin(x) or sin^(-1)(x), with respect to x, can be found using calculus. The derivative of arcsin(x) is given by:

d/dx(arcsin(x)) = 1 / √(1 – x^2)

This result can be derived using the chain rule and trigonometric identities. If you have a function f(x) that involves arcsin(x), you can use this derivative to find the derivative of f(x) with respect to x.

For example, if you have a function f(x) = arcsin(2x), you can find its derivative as follows:

d/dx(f(x)) = d/dx(arcsin(2x)) = (1 / √(1 – (2x)^2)) * 2 = 2 / √(1 – 4x^2)

So, the derivative of f(x) = arcsin(2x) is 2 / √(1 – 4x^2).

Inverse Sine Value Table

The inverse sine function, also known as arcsin, gives you the angle (in radians or degrees) whose sine is equal to a given value.

Here is a table of some common inverse sine values:

Degrees (°)	Radians (rad)	arcsin Value
0°	0	0
30°	π/6 (0.5236)	0.5
45°	π/4 (0.7854)	0.7071
60°	π/3 (1.0472)	0.8660
90°	π/2 (1.5708)	1.0

You can easily calculate other values using trigonometric calculators or software. Remember that the arcsin function returns values in the range of [-1, 1], and the result will be negative for angles in the 3rd and 4th quadrants.

Properties of Inverse Sine Function

The inverse sine function, denoted as “arcsin” or “asin,” is the inverse of the sine function. It takes an input in the range [-1, 1] and returns an angle in the range [-π/2, π/2] or, in degrees, [-90°, 90°] that corresponds to the input’s sine value. Here are some key properties of the inverse sine function:

Domain and Range:
- Domain: The domain of the inverse sine function is [-1, 1]. It takes real numbers within this interval as its input.
- Range: The range of the inverse sine function is [-π/2, π/2] in radians or [-90°, 90°] in degrees.
Notation: The inverse sine function is often denoted as arcsin, asin, or sin^(-1).
Inverse Relationship: The arcsin function is the inverse of the sine function. If you take the sine of an angle and then apply the arcsin function, you will return to the original angle. In other words, for all x in the domain of [-1, 1], arcsin(sin(x)) = x.
Symmetry: The arcsin function is odd, meaning that arcsin(-x) = -arcsin(x) for all x in its domain.
Trigonometric Identity: The inverse sine function is related to the sine function through the identity: sin(arcsin(x)) = x for all x in its domain.
Range Restriction: The range of the arcsin function is limited to [-π/2, π/2] or [-90°, 90°]. This restriction is necessary to make it a true inverse function, as it ensures a unique output for each input within its domain.
Principal Value: The principal value of the arcsin function is the value within its range that corresponds to the sine value. It is typically the value in the interval [-π/2, π/2] for radians or [-90°, 90°] for degrees.
Inverse Trigonometric Identities: The inverse sine function is one of the trigonometric functions that have various identities relating it to other trigonometric functions. For example:
- cos(arcsin(x)) = √(1 – x^2)
- tan(arcsin(x)) = x / √(1 – x^2)
Inverse Trigonometric Function Properties: The inverse sine function shares some general properties with other inverse trigonometric functions, such as arcsin(-x) = -arcsin(x), and arcsin(x) + arcsin(√(1 – x^2)) = π/2 or 90°, which can be useful in solving trigonometric equations.
Periodicity: The inverse sine function is not periodic; its output values within the range of [-π/2, π/2] or [-90°, 90°] do not repeat in a periodic fashion as the input values change.

READ What Is A Disjoint Set, How Do You Prove A Set Is Disjoint?

These properties make the inverse sine function a fundamental tool in trigonometry, calculus, and various fields of science and engineering where angles and sine values are involved.

Some Solved Examples on Inverse Sine Function

Here are some solved examples involving the inverse sine function, denoted as “arcsin” or “sin^(-1)”.

Example 1: Find the value of arcsin(1).

Solution: The inverse sine function returns an angle between -π/2 and π/2 whose sine is equal to the given value. Since the sine of 1 is equal to 1, we have:

arcsin(1) = π/2

Example 2: Find the value of sin(arcsin(0.6)).

Solution: First, you find the angle whose sine is 0.6 by using the inverse sine function:

arcsin(0.6) ≈ 0.6435 radians

Then, you find the sine of this angle:

sin(0.6435) ≈ 0.6

So, sin(arcsin(0.6)) = 0.6.

Example 3: Solve for x: sin(arcsin(x)) = 0.8.

Solution: We want to find the value of x when sin(arcsin(x)) equals 0.8. The inverse sine function and sine function cancel each other out, so:

sin(arcsin(x)) = x

Now, we have:

x = 0.8

Example 4: Find the value of arcsin(√3/2).

Solution: The sine of π/6 radians (or 30 degrees) is √3/2. Therefore:

arcsin(√3/2) = π/6

Example 5: Find the value of sin(arcsin(-0.7071)).

Solution: First, you find the angle whose sine is approximately -0.7071:

arcsin(-0.7071) ≈ -π/4

Then, you find the sine of this angle:

sin(-π/4) ≈ -0.7071

So, sin(arcsin(-0.7071)) ≈ -0.7071.

These examples illustrate how to use the inverse sine function to find angles or values when working with trigonometric equations and identities involving sine and arcsin.

Thank you so much for taking the time to read the article titled What is Inverse Sine Function? written by Math Hello Kitty. Your support means a lot to us! We are glad that you found this article useful. If you have any feedback or thoughts, we would love to hear from you. Don’t forget to leave a comment and review on our website to help introduce it to others. Once again, we sincerely appreciate your support and thank you for being a valued reader!

Source: Math Hello Kitty
Categories: Math

What is Inverse Sine Function?