Sine Calculator
Calculate sin(x) for any angle or find the angle from a sine value using arcsin. Features unit circle and sine wave visualizations.
Result
sin(30)
0.500000
Exact: 1/2
Calculation Mode
Input
Enter angle in degrees
Unit Circle Visualization
On the unit circle, sin(30.0) is the y-coordinate of the point: 0.500000
Related Trigonometric Values
sin(30)
0.500000
cos(30)
0.866025
tan(30)
0.577350
csc(30) = 1/sin
2.000000
Cofunction Identity: sin(30) = cos(60)
Sine Wave Graph
Amplitude
1
Period
2pi
Phase Shift
0
Vertical Shift
0
Equation:
y = sin((x))
Special Sine Values
| Angle (deg) | Radians | sin | Exact Value |
|---|---|---|---|
| 0 | 0 | 0.0000 | 0 |
| 30 | pi/6 | 0.5000 | 1/2 |
| 45 | pi/4 | 0.7071 | sqrt(2)/2 |
| 60 | pi/3 | 0.8660 | sqrt(3)/2 |
| 90 | pi/2 | 1.0000 | 1 |
| 120 | 2pi/3 | 0.8660 | sqrt(3)/2 |
| 135 | 3pi/4 | 0.7071 | sqrt(2)/2 |
| 150 | 5pi/6 | 0.5000 | 1/2 |
| 180 | pi | 0.0000 | 0 |
Sine Function Properties
Domain
All real numbers (-inf, inf)
Range
[-1, 1]
Period
360 degrees (2pi radians)
Odd/Even
Odd function: sin(-x) = -sin(x)
Zeros
x = n*180 (n*pi), where n is integer
Maximum/Minimum
Max = 1 at 90, Min = -1 at 270
Applications of Sine
Sound Waves
Pure tones are modeled by sine waves. The frequency determines pitch, amplitude determines volume.
Harmonic Motion
Pendulums, springs, and oscillations follow sinusoidal patterns: x(t) = A*sin(omega*t + phi)
AC Electricity
Alternating current is sinusoidal: V(t) = V_max * sin(2*pi*f*t)
Navigation
Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) is used in triangulation and surveying.
Result
sin(30)
0.500000
Try These Examples
Quick-start with common scenarios
Practice Problems
Test your skills with practice problems
Practice with 3 problems to test your understanding.
?What is Sine?
Sine (sin) is a trigonometric function that gives the ratio of the opposite side to the hypotenuse in a right triangle: sin(angle) = opposite/hypotenuse. On the unit circle, sine equals the y-coordinate. Common values: sin(0)=0, sin(30)=0.5, sin(45)=sqrt(2)/2, sin(60)=sqrt(3)/2, sin(90)=1. The inverse sine (arcsin) finds the angle from a sine value.
About the Sine Function
The sine function is one of the fundamental trigonometric functions. For an angle in a right triangle, sine equals the length of the opposite side divided by the hypotenuse. In the unit circle, sine represents the y-coordinate of a point on the circle. Sine oscillates between -1 and 1 and is used extensively in physics for modeling waves, oscillations, and periodic phenomena.
Key Facts
- sin(x) = opposite / hypotenuse in a right triangle
- On the unit circle, sin(x) = y-coordinate of the point
- Range: sine values are always between -1 and 1
- Period: sine repeats every 360 degrees (2pi radians)
- sin(0) = 0, sin(90) = 1, sin(180) = 0, sin(270) = -1
- arcsin(x) returns angle in range [-90, 90] or [-pi/2, pi/2]
- Sine is positive in quadrants I and II (0-180 degrees)
- sin(x) = cos(90 - x) - coffunction identity
Frequently Asked Questions
Sine is a trigonometric function that, for an angle in a right triangle, equals the ratio of the opposite side to the hypotenuse. On the unit circle, sin(angle) equals the y-coordinate of the point at that angle. Sine oscillates between -1 and 1.
sin(30) = 0.5 or 1/2. This is one of the special angles. The exact value comes from the 30-60-90 triangle where the side opposite to 30 is half the hypotenuse.
Arcsin (also written as sin inverse or asin) returns the angle whose sine equals a given value. For example, arcsin(0.5) = 30 because sin(30) = 0.5. The result is always between -90 and 90 degrees.
Sine is positive in quadrants I and II (0 to 180 degrees) where the y-coordinate is positive. Sine is negative in quadrants III and IV (180 to 360 degrees) where the y-coordinate is negative.
The sine function has a period of 360 degrees (or 2pi radians). This means sin(x) = sin(x + 360) for any angle x. The function completes one full oscillation every 360 degrees.
Last updated: 2026-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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Result
sin(30)
0.500000
Exact: 1/2