What is Inverse Sine?
The inverse sine (also called arcsine or sin-1) is the inverse of the sine function. It finds the angle that produces a particular sine value.
arcsin(x) = y if and only if x = sin(y)
where x ∈ [-1, 1]
Domain and Range of Inverse Sine
Domain of arcsin
The domain of the inverse sine is the interval [-1, 1]. This is because:
- The range of the sine function is [-1, 1]
- The range of a function becomes the domain of its inverse
- You can only calculate arcsin for values between -1 and 1
Range of arcsin
The range of the inverse sine is:
- In radians: [-π/2, π/2] or approximately [-1.571, 1.571]
- In degrees: [-90°, 90°]
This range is chosen because the sine function is periodic (many-to-one). To invert it, we restrict sine to the interval [-π/2, π/2] where it is one-to-one.
How to Calculate Inverse Sine
Using This Calculator
- Enter the sine value (x) in the input field
- The value must be between -1 and 1 (inclusive)
- The calculator instantly displays arcsin(x) in both radians and degrees
Bonus: This calculator works in reverse too! Switch to the "Sine" tab to calculate the sine of an angle.
Common Inverse Sine Values
| x | arcsin(x) (radians) | arcsin(x) (degrees) |
|---|---|---|
| -1 | -π/2 ≈ -1.571 | -90° |
| -√3/2 ≈ -0.866 | -π/3 ≈ -1.047 | -60° |
| -√2/2 ≈ -0.707 | -π/4 ≈ -0.785 | -45° |
| -1/2 = -0.5 | -π/6 ≈ -0.524 | -30° |
| 0 | 0 | 0° |
| 1/2 = 0.5 | π/6 ≈ 0.524 | 30° |
| √2/2 ≈ 0.707 | π/4 ≈ 0.785 | 45° |
| √3/2 ≈ 0.866 | π/3 ≈ 1.047 | 60° |
| 1 | π/2 ≈ 1.571 | 90° |
Practical Examples
Example 1: arcsin(0.5)
Question: What angle has a sine of 0.5?
- Solution: arcsin(0.5) = π/6 ≈ 0.524 radians = 30°
- Explanation: In a right triangle, when the opposite side is half the hypotenuse, the angle is 30°
Example 2: arcsin(1)
Question: What angle has a sine of 1?
- Solution: arcsin(1) = π/2 ≈ 1.571 radians = 90°
- Explanation: When the opposite side equals the hypotenuse, the angle is 90° (a right angle)
Example 3: arcsin(-0.707)
Question: What angle has a sine of approximately -0.707?
- Solution: arcsin(-0.707) ≈ -π/4 ≈ -0.785 radians = -45°
- Explanation: The negative value indicates an angle below the x-axis
Inverse Sine Graph
The graph of y = arcsin(x) has the following properties:
- Domain: x ∈ [-1, 1]
- Range: y ∈ [-π/2, π/2] or [-90°, 90°]
- Increasing function: As x increases, y increases
- Passes through origin: arcsin(0) = 0
- Symmetric about origin: arcsin(-x) = -arcsin(x)
Relationship with Other Inverse Trig Functions
Inverse Sine vs Inverse Cosine
arcsin(x) + arccos(x) = π/2 (or 90°)
This relationship holds for all x ∈ [-1, 1]
Using Pythagorean Identity
If θ = arcsin(x), then:
- sin(θ) = x
- cos(θ) = √(1 - x²)
- tan(θ) = x / √(1 - x²)
Frequently Asked Questions
How do I calculate the inverse sine of one half?
- Sketch a right-angled triangle
- Recall that sine is the ratio of the opposite side to the hypotenuse
- We're looking for the angle where the hypotenuse is twice as long as the opposite side
- From geometry, this angle is 30° (or π/6 radians)
- Therefore, arcsin(0.5) = 30°
What is the difference between sin-1 and 1/sin?
sin-1(x) means the inverse sine function (arcsin), while 1/sin(x) means the reciprocal of sine (which is cosecant or csc). These are completely different:
- sin-1(0.5) = 30° - the angle whose sine is 0.5
- 1/sin(30°) = 2 - the reciprocal of the sine of 30°
Why is the range of arcsin limited to [-90°, 90°]?
The sine function is periodic and repeats every 360°. Without limiting the range, arcsin would have infinite possible answers. We restrict the range to [-90°, 90°] to ensure each input has exactly one output, making arcsin a proper function.
Can I calculate arcsin for values outside [-1, 1]?
No, arcsin is only defined for values between -1 and 1 (inclusive). Values outside this range have no real solution because the sine of any real angle always produces a value between -1 and 1.
What is arcsin(0)?
arcsin(0) = 0° (or 0 radians). This is because sin(0°) = 0. The angle of 0° has a sine value of 0.