arcsin Calculator

Consider the right-angled triangle below.

In this triangle, the right angle is at B. Sine of the angle y is given by

$$sin\, y\, =\, {opposite \over hypotenuse} \,=\,{AB\over AC} = x$$

To find the angle y when the ratio AB/AC is known, we use the following formula for arcsine

$$y\, =\, sin^{-1}{opposite \over hypotenuse} \,=\,sin^{-1} x$$

The expression y = sin^-1⁡x, means that sin y = x , when -π/2 ≤ y ≤ π/2 and -1 ≤ x ≤ +1.

Arcsine(x)=y is defined as the set of all angles whose sine is x. It should be noted here that the -1 is not an exponent and hence, sin^-1⁡x ≠ 1/sin⁡x .

Given below are some properties of the arcsine function.

Angle – The angle is the parameter which is calculated when sin inverse (or arcsine) of the angle is known.

Domain – The arcsine of the parameter whose sine is x. This value goes from -1 ≤ x ≤ 1.

Range – The angles for which we can find sin^(-1)⁡x. This value is -π/2≤ y ≤π/2 radians or -90 ≤ y ≤ +90 degrees.

Given below is a graph of the arcsine function.

In the above graph, the y-axis denotes the angle in radians, and the x-axis denotes arcsine of that angle. The red curve denotes the arcsine function. For any value of x between -1 and +1, the arcsine function returns a value between -π/2 and π/2 radians (or -90⁰ to 90⁰). For example, when x=1, y=arcsine(x) returns y= π/2 radians. This means that sin(π/2) = 1.

If the arcsine graph is compared to the sine graph, one would see that the arcsine graph can be obtained from the sine graph by simply interchanging the horizontal and vertical axes.

Given below are some commonly used values of arcsine(x).

The value of arcsine(x) can be positive or negative, depending on the value of x. For example, arcsine(-1/2) is negative, whereas arcsine(1/2) is positive. This is because sin^-1⁡(-x)=-sin^-1⁡x.

This has been illustrated below.

In this figure, the x-axis denotes the arcsine values, whereas the y-axis denotes the angle θ. Arcsine(x) will be positive when 0≤ θ ≤ π/2 (in Quadrant 1), and negative when -π/2 ≤ θ < 0 (in Quadrant 4).

We can also note that

sin⁡(sin^-1⁡(x))=x when -1 ≤ x ≤ +1.

For example, sin⁡(sin^-1⁡(1/2)) = sin⁡ 30⁰ = 1/2.

Similarly, sin^-1⁡⁡(sin⁡ x) = x when -π/2 ≤ x ≤ +π/2.

For example, sin^-1⁡⁡(sin⁡ π/4) = sin^-1⁡⁡⁡(1/√2) = π/4.

If we need to find the arcsine of the sine of an angle which does not lie between -π/2 and π/2, the correct angle can be found by either adding or subtracting 2π radians until we get an angle in the range of -π/2 to +π/2, which is the range of the arcsine function. This is because 2π is the period of the sin function.

From a right-angled triangle, the arcsine formula can be computed as

$$y = sin^{-1}{AB \over AC} = sin^{-1} \,x$$

where AB is the opposite side and AC is the hypotenuse, and y is the angle which can be calculated.

Like sin(x), there is no well-known Taylor series approximation for the arcsine function. However, a series expansion can be obtained using binomial expansion or integration.

The formula thus obtained, where -1 < x < 1, is given below.

OR

Trigonometric functions as well as the inverse trigonometric functions are used widely in engineering, geometry and physics. They are also used to solve trigonometric equations describing real-life problems.

In this section, we consider some example problems involving the use of the arcsine formula.

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