Free Sine Calculator

The Sine Calculator is a free online trigonometry calculator that provides instant sine values or angle conversions. Enter any angle in degrees or radians to compute its sine, or supply a sine value to obtain the corresponding angle (within the principal range $-90^\circ$ to $90^\circ$). This tool serves as a practical Sine Function Calculator for students, engineers, and anyone performing trigonometric work.

Sine Definition: Right Triangle and Unit Circle

Sine is one of the three fundamental trigonometric functions, alongside cosine and tangent. Two standard definitions explain its meaning:

• Right‑Triangle Model: For an acute angle $\alpha$ (between $0^\circ$ and $90^\circ$), $\sin(\alpha)$ equals the length of the side opposite $\alpha$ divided by the length of the hypotenuse. The ratio holds true for any right triangle because of similarity.

• Unit‑Circle Model: For any angle $\theta$ measured counterclockwise from the positive $x$-axis, $\sin(\theta)$ is the $y$-coordinate of the point where the terminal side meets the unit circle (a circle of radius $1$ centered at the origin). This definition extends sine to all real angles and underpins periodic analysis.

Key Properties of the Sine Function

• Domain: All real numbers.
• Range: $[-1, 1]$ — the output never goes below $-1$ or above $1$.
• Periodicity: The function repeats every $2\pi$ radians (360°): $\sin(\theta + 2\pi) = \sin(\theta)$.
• Odd Symmetry: $\sin(-\theta) = -\sin(\theta)$.
• Zeros (Roots): $\sin(k\pi) = 0$ for every integer $k$.
• Maximum: $1$ at $\theta = \frac{\pi}{2} + 2k\pi$.
• Minimum: $-1$ at $\theta = \frac{3\pi}{2} + 2k\pi$.

The Sine Wave (Sinusoid)

A sinusoidal signal is expressed by

$$y = A \sin(2\pi f t + \varphi) \quad \text{or} \quad y = A \sin(\omega t + \varphi),$$

where:

• $A$ — amplitude (peak deviation from zero),
• $f$ — ordinary frequency (cycles per second),
• $\omega = 2\pi f$ — angular frequency (radians per second),
• $\varphi$ — phase shift (radians).

Sine waves describe natural phenomena such as sound, light, and alternating currents, and are central to physics, engineering, and signal processing.

Selected Sine Values

Several common angles yield sine values that are easy to express exactly and as decimals.

Angle (°)	Angle (rad)	$\sin$ (exact)	$\sin$ (approx.)
$0$	$0$	$0$	$0$
$15$	$\frac{\pi}{12}$	$\frac{\sqrt{6} - \sqrt{2}}{4}$	$0.2588190451$
$30$	$\frac{\pi}{6}$	$\frac{1}{2}$	$0.5$
$45$	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$0.7071067812$
$60$	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$0.8660254038$
$75$	$\frac{5\pi}{12}$	$\frac{\sqrt{6} + \sqrt{2}}{4}$	$0.9659258263$
$90$	$\frac{\pi}{2}$	$1$	$1$
$120$	$\frac{2\pi}{3}$	$\frac{\sqrt{3}}{2}$	$0.8660254038$
$135$	$\frac{3\pi}{4}$	$\frac{\sqrt{2}}{2}$	$0.7071067812$
$150$	$\frac{5\pi}{6}$	$\frac{1}{2}$	$0.5$
$180$	$\pi$	$0$	$0$

Values for angles outside this table can be derived using periodic shifts and symmetry (e.g., $\sin(\alpha + 360^\circ) = \sin(\alpha)$).

Sine by Quadrant

The sign and monotonic behavior of sine vary across the four quadrants:

Quadrant	Degrees	Radians	Sign	Monotonicity
1st	$0^\circ < \theta < 90^\circ$	$0 < \theta < \frac{\pi}{2}$	$+$	Increasing
2nd	$90^\circ < \theta < 180^\circ$	$\frac{\pi}{2} < \theta < \pi$	$+$	Decreasing
3rd	$180^\circ < \theta < 270^\circ$	$\pi < \theta < \frac{3\pi}{2}$	$-$	Decreasing
4th	$270^\circ < \theta < 360^\circ$	$\frac{3\pi}{2} < \theta < 2\pi$	$-$	Increasing

Critical points: $\sin(0^\circ) = 0$, $\sin(90^\circ) = 1$, $\sin(180^\circ) = 0$, $\sin(270^\circ) = -1$. Due to periodicity, these patterns repeat every $360^\circ$.

How to Use the Sine Calculator

This Sine of Angle Calculator is straightforward:

1. Find sine from an angle: Choose degrees or radians, then type the angle. For example, entering $40^\circ$ yields $\sin(40^\circ) \approx 0.6428$.
2. Find angle from a sine: Input a sine value (e.g., $0.7$) and the Sine Calculator returns the principal angle in $[-90^\circ, 90^\circ]$—here, $44.43^\circ$. Use periodicity to find other angles: $\sin(44.43^\circ + 360^\circ) = \sin(404.43^\circ) = 0.7$.

The tool works as both a direct sine computer and an inverse sine finder, making it a versatile Trigonometry Calculator.

Further Applications

Understanding sine is essential for solving triangles (law of sines), modeling periodic phenomena, and analyzing waves. While this calculator handles core sine tasks, related resources can assist with cosine, tangent, and trigonometric identities.