Sin Calculator
Need the sine of 37° or sin(1.2 radians) without scrolling through a textbook? The sin calculator above computes the exact value instantly – just enter the angle and pick the unit.
What Is the Sine Function?
The sine (abbreviated sin) is a trigonometric function that relates an angle to the ratio of two sides of a right triangle. For an angle θ in a right triangle:
sin(θ) = opposite side ÷ hypotenuse
On the unit circle – a circle with radius 1 centered at the origin – sin(θ) equals the y-coordinate of the point where the angle’s terminal side intersects the circle.
Key properties of the sine function:
- Domain: all real numbers (−∞ to +∞)
- Range: −1 to 1
- Period: 2π radians (360°)
- Odd function: sin(−θ) = −sin(θ)
How Do You Calculate Sin of an Angle?
The calculator above handles the computation directly: enter an angle value, select degrees or radians, and read the result. Here is what happens behind the scenes.
For angles in degrees, the value is first converted to radians:
radians = degrees × π / 180
Then the sine is evaluated using the series expansion or a built-in algorithm:
sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + …
This Taylor series converges for all real numbers. In practice, calculators use optimized algorithms like CORDIC or polynomial approximations for speed and precision.
For manual work with common angles, exact values are faster than any formula – see the table below.
Sin Values Table for Common Angles
| Angle (degrees) | Angle (radians) | sin(θ) |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.5 |
| 45° | π/4 | ≈ 0.7071 |
| 60° | π/3 | ≈ 0.8660 |
| 90° | π/2 | 1 |
| 120° | 2π/3 | ≈ 0.8660 |
| 135° | 3π/4 | ≈ 0.7071 |
| 150° | 5π/6 | 0.5 |
| 180° | π | 0 |
| 270° | 3π/2 | −1 |
| 360° | 2π | 0 |
Notice the symmetry: sin(180° − θ) = sin(θ), and sin(180° + θ) = −sin(θ). These identities let you find the sine of any angle by relating it to a first-quadrant angle (0° to 90°).
What Is the Difference Between Degrees and Radians?
Degrees and radians are two units for measuring angles, and mixing them up is the most common source of wrong sine values.
- Degree: divides a full circle into 360 equal parts. 1° = π/180 radians.
- Radian: based on arc length. One radian is the angle that subtends an arc equal to the radius. A full circle = 2π ≈ 6.2832 radians.
Most mathematical formulas, programming languages, and the Taylor series assume radians. If your angle is in degrees, always convert first:
sin(30°) = sin(30 × π/180) = sin(π/6) = 0.5
Entering sin(30) in radian mode returns ≈ −0.9880 – a dead giveaway that the unit was wrong.
Key Sine Formulas and Identities
These identities appear constantly in algebra, calculus, and physics:
- Pythagorean identity: sin²θ + cos²θ = 1
- Double angle: sin(2θ) = 2 sin θ cos θ
- Sum formula: sin(α + β) = sin α cos β + cos α sin β
- Difference formula: sin(α − β) = sin α cos β − cos α sin β
- Half angle: sin(θ/2) = ±√((1 − cos θ) / 2)
From the Pythagorean identity, you can derive cos θ if you know sin θ:
cos θ = ±√(1 − sin²θ)
The sign depends on the quadrant of the angle.
Where Is the Sine Function Used?
Sine is not just a textbook concept – it models any periodic, oscillating phenomenon:
- Physics: simple harmonic motion (pendulums, springs), sound waves, and electromagnetic waves all follow sinusoidal patterns.
- Electrical engineering: AC voltage is V(t) = V_peak × sin(ωt), where ω is the angular frequency.
- Navigation and surveying: triangulation relies on the sine rule: a/sin A = b/sin B = c/sin C.
- Computer graphics: sine waves generate smooth oscillations, procedural textures, and camera movements.
- Signal processing: Fourier analysis decomposes any signal into a sum of sine and cosine components.
Whether you are solving a homework problem, designing a circuit, or debugging a signal, the sin calculator above gives you an instant result without switching contexts.