Sine Calculator
Enter an angle to get its sine (plus cosine, tangent, and the angle expressed in every unit). Switch to inverse mode to find the angle from a known sine value. Results snap to exact textbook values for common angles.
Formula
Worked example
For θ = 30°: convert to radians (30 × π/180 = π/6 ≈ 0.5236 rad), then sin(π/6) = 0.5, cos(π/6) = √3/2 ≈ 0.8660, and tan(π/6) = 1/√3 ≈ 0.5774, exactly the classic 30-60-90 triangle ratios.
What the sine of an angle means
In a right triangle, the sine of an acute angle is the ratio of the side opposite that angle to the hypotenuse. Because the hypotenuse is always the longest side, this ratio can never exceed one, which is why sin(θ) is forever trapped between -1 and 1. The unit circle generalizes this to any angle: as a point travels around a circle of radius one, its height above the horizontal axis is exactly the sine of the rotation angle. That is why sine is not just a triangle tool but the mathematical language of any repeating, wave-like phenomenon: sound waves, AC voltage, tidal cycles, mechanical vibrations, and light all follow sinusoidal patterns.
Degrees, radians, and pi notation
The same physical angle has two common numerical names. Degrees divide a full turn into 360 equal parts, a convention inherited from ancient Babylonian astronomy. Radians measure the angle by arc length on a unit circle, so a full turn equals 2π. Programming languages, spreadsheets, and scientific calculators almost always compute sine internally in radians, which is why a value entered as degrees must first be multiplied by π/180. A third notation writes the angle as a multiple of π directly: 30° becomes π/6, 90° becomes π/2, and 180° becomes π. This calculator shows all three representations at once so you never have to convert by hand.
Quadrant rules and sign of sine
A circle is divided into four quadrants by the x and y axes. In Quadrant I (0° to 90°) both sine and cosine are positive. In Quadrant II (90° to 180°) sine remains positive but cosine turns negative, which is why sin(150°) = sin(30°) = 0.5. In Quadrant III (180° to 270°) both are negative. In Quadrant IV (270° to 360°) sine is negative but cosine returns to positive. The mnemonic "All Students Take Calculus" (A-S-T-C) names the positive function in each quadrant starting at I: All, Sine, Tangent, Cosine. This calculator always reports the quadrant so you can reason about sign without drawing a diagram.
Inverse sine: finding the angle from a known sine value
When you know the sine and need the angle, use the inverse sine function, written arcsin or sin-1. Because sine is not a one-to-one function (sin(30°) = sin(150°) = 0.5), arcsin returns only the principal value, the one angle in the range -90° to 90° with that sine. A second angle, the supplement (180° minus the principal value), also has the same sine in the range 0° to 180°. Beyond those two, infinitely many angles share the same sine by adding or subtracting multiples of 360°. Switch this calculator to inverse mode, enter the sine value, and both the principal angle and its supplement are shown instantly.
Cosine, tangent, and the Pythagorean identity
Once you have sin(θ), two companion values follow immediately. The cosine is the horizontal projection of the same unit-circle point, giving the ratio of the adjacent side to the hypotenuse. The tangent is their ratio: tan(θ) = sin(θ) / cos(θ), equal to the ratio of opposite to adjacent. The three are linked by the Pythagorean identity: sin²(θ) + cos²(θ) = 1 for every angle without exception. Knowing any one of them (together with the quadrant) lets you recover the other two. This calculator shows all three so you have a complete picture from a single input.
Common uses: triangles, waves, and engineering
The sine function appears in virtually every branch of science and engineering. In surveying and navigation, the law of sines (a/sin A = b/sin B = c/sin C) links the sides of any triangle to the sines of the opposite angles, allowing unknown sides or angles to be found when a complete solution is needed. In physics and electrical engineering, alternating current is described by V = V₀ sin(2πft), where f is frequency and t is time. In audio, sine waves are the pure tones from which all complex sounds can be built (Fourier analysis). Understanding the sine function and being able to compute it instantly is therefore foundational across many disciplines.
Sine, cosine and tangent of common angles
| Angle (deg) | Angle (rad) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 = 0.5000 | √3/2 = 0.8660 | 1/√3 = 0.5774 |
| 45° | π/4 | √2/2 = 0.7071 | √2/2 = 0.7071 | 1 |
| 60° | π/3 | √3/2 = 0.8660 | 1/2 = 0.5000 | √3 = 1.7321 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 = 0.8660 | -1/2 = -0.5000 | -√3 = -1.7321 |
| 135° | 3π/4 | √2/2 = 0.7071 | -√2/2 = -0.7071 | -1 |
| 150° | 5π/6 | 1/2 = 0.5000 | -√3/2 = -0.8660 | -1/√3 = -0.5774 |
| 180° | π | 0 | -1 | 0 |
| 210° | 7π/6 | -1/2 = -0.5000 | -√3/2 = -0.8660 | 1/√3 = 0.5774 |
| 270° | 3π/2 | -1 | 0 | undefined |
| 330° | 11π/6 | -1/2 = -0.5000 | √3/2 = 0.8660 | -1/√3 = -0.5774 |
| 360° | 2π | 0 | 1 | 0 |
Exact and decimal values for frequently used angles across all four quadrants.
Frequently asked questions
Why does my calculator give a different sine value than yours?
The most common cause is a unit mismatch. If your device is set to radians but you entered a value intended as degrees (or vice versa), the result will be completely different: sin(30) in radians is about 0.9880, not 0.5. Check that the unit selector here and the DEG/RAD mode on your own device both agree before comparing results.
Can the sine of an angle ever be greater than 1?
No. Sine is the ratio of the opposite side to the hypotenuse in a right triangle, and the hypotenuse is always the longest side, so the ratio can never exceed 1. Across all angles, sin(θ) is bounded between -1 and 1 inclusive. Values outside this range indicate a calculation error or a unit problem.
How do I find the angle if I already know its sine?
Use the inverse sine (arcsin). Switch this calculator to inverse mode, enter the sine value, and it returns the principal angle in the range -90° to 90° and the supplement (180° minus the principal). For example, arcsin(0.5) gives 30° as the principal angle and 150° as the supplement: both have a sine of exactly 0.5.
What is the Pythagorean identity and why does it matter?
The identity sin²(θ) + cos²(θ) = 1 holds for every angle without exception. It follows directly from the Pythagorean theorem applied to a unit circle: the horizontal and vertical projections of the radius are cos(θ) and sin(θ), and they form the legs of a right triangle with hypotenuse 1. The identity lets you find cos(θ) from sin(θ) (or vice versa) whenever you know the quadrant: cos(θ) = ±√(1 - sin²(θ)).
What is the difference between sin(x), arcsin(x) and sin-1(x)?
They mean the same thing two different ways. sin(x) takes an angle and returns a ratio. arcsin(x) and sin-1(x) are both names for the inverse function that takes a ratio and returns an angle. The exponent notation sin-1 is common on calculator buttons and in textbooks, but it can be confused with 1/sin(x) (which is cosecant). arcsin(x) is unambiguous and preferred in most modern writing.
How is sine used in real life?
Sine appears in almost every physical wave. Alternating current voltage follows V = V₀ sin(2πft), sound is described as a sum of sine waves (Fourier series), and ocean tides are modeled with sinusoidal functions. In navigation and surveying the law of sines links a triangle's sides and angles. In architecture and engineering, sine and cosine resolve forces into horizontal and vertical components. Anywhere there is oscillation, rotation, or a right-triangle relationship, sine is involved.