Inverse Sine Calculator

This inverse sine calculator will help you quickly calculating the value of the inverse of the sine function. In the article below, we discuss what it means to invert the sine function and what the graph of the sine inverse looks like. In particular, we'll explain the domain and range of sin inverse. Let's go!

The inverse sine is, as its name suggests, the inverse of the sine function. That is, inverse sine finds the angle that produces a particular value of sine. The common notation of this function is arcsin:

arcsin(x) = y if and only if x = sin(y),

where x ∈ [-1, 1].

Wondering where this last condition comes from? Let's move on to discussing the domain and range of inverse sine.

We can calculate the inverse sine only for values between -1 and 1. In other words, the domain of the inverse sine is the interval [-1, 1].

This is because the range of a function becomes the domain of its inverse function. Since the sine value assumes values between -1 and 1, this interval is the domain of the inverse sine.

The range of inverse sine is the interval in radians [-π/2, π/2], that is [-90 deg, 90 deg]. To understand why this is the case, recall that the sine function is periodic, in particular, it is many-to-one. We cannot invert it as it is: first, we have to restrict sine to an interval where it is one-to-one. Most often, we choose the interval [-π/2, π/2] as the domain of the restricted sine. It is, therefore the range of the inverse sine.

We can summarize the above considerations by taking a look at the inverse sine graph:

This tool is very straightforward to use. You only need to input the argument x and the value of arcsin(x) will appear. Just remember what we've explained about the domain of inverse sine: the value you enter must satisfy the condition -1 ≤ x ≤ 1.

Tip: as the vast majority of Omni tools, this inverse sine calculator can work in reverse, that is, you can use it as a sine calculator!

Enjoyed using out inverse sine calculator? We've built a whole bunch of tools related to inverse trig functions, each highlighting the topic from a slightly different angle ;) Here they are:

• Arcsin calculator;
• Sin-1 calculator;
• Cos-1 calculator; and
• Arccos calculator.

To determine the inverse sine of ½:

1. Sketch a right-angled triangle.
2. Recall that the sine is the ratio of the opposite side to the hypotenuse.
3. We're looking for the angle for which the hypotenuse is twice as long as the opposite side.
4. We recall from geometry class that this angle must be 30°.
5. If your memory refuses to retain stuff taught in geometry class, use an online sin inverse calculator to find the answer!