Arcsine Calculator
The Arc Sine (or inverse sine) function is the inverse of the sine function, which means it helps you find the angle whose sine is a given value. It is commonly written as $$\arcsin(x)$$ or $$\sin^{-1}(x)$$.
What is Arc Sine?
The arc sine function answers the question: “What angle has a sine of x?” For example, if the sine of 30° is 0.5, then $$\arcsin(0.5)$$ equals 30°. Note that the arc sine function is only defined for values of $$x$$ in the range $$[-1, 1]$$, because sine values cannot exceed this range.
For example:
- $$\arcsin(0.5)$$ = 30°, because $$\sin(30^\circ) = 0.5$$.
- $$\arcsin(-0.5)$$ = -30°, because $$\sin(-30^\circ) = -0.5$$.
- $$\arcsin(1)$$ = 90°, because $$\sin(90^\circ) = 1$$.
How Does Arc Sine Work?
The arc sine function is a bijection between the interval $$[-1, 1]$$ and the interval $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$ in radians (or -90° to 90° in degrees). When calculating the arc sine, it will always return an angle in this range, called the principal value.
For example, if the sine of both 30° and 150° is 0.5, the arc sine function will return 30°, because it falls within the principal range.
Properties of Arc Sine
- $$\arcsin(0)$$ = 0 because $$\sin(0) = 0$$.
- For $$x = 1$$, $$\arcsin(x)$$ = $$\frac{\pi}{2}$$ (90°).
- For $$x = -1$$, $$\arcsin(x)$$ = $$-\frac{\pi}{2}$$ (-90°).
- The arc sine function is increasing, meaning that as $$x$$ increases, so does $$\arcsin(x)$$.
Why Use the Arc Sine Function?
The arc sine function is widely used in trigonometry, physics, and engineering. Some applications include:
- Solving triangles: It helps find angles when the lengths of the opposite side and hypotenuse are known.
- Signal processing: Used in calculating phase angles for sinusoidal waveforms.
- 3D graphics: Determines angles for rotations and transformations in 3D space.
Using our arc sine calculator, you can very easily calculate the inverse sine of any valid input. Whether you're working with degrees or radians, the tool will give you precise results, simplifying your calculations and saving time.
