With this sine calculator, you can find the sine value in the blink of an eye — all you need to do is type the angle in degrees or radians (gradians work too). The calculator also works the other way round: put the value of sine into the proper box, and we'll calculate the angle for you. As a result, you'll get the angle from the <−90°, 90°> range.
Scroll down to understand what a sine is and to find the sine definition, as well as simple examples and the sine graph. You'll also find a handy table with values of sine for basic angles, such as sin(0), sin(30°), and many more.
What Is the Sine Function? Sine Definition
Sine is one of the three most common trigonometric functions (the others are cosine and tangent, as well as secant, cosecant, and cotangent). The abbreviation of sine is sin, e.g., sin(30°). The most common and well-known sine definition is based on the right-angled triangle.
Let's start with the nomenclature of the triangle sides. As the picture on the right shows, we can name the sides of a right triangle as:
- Adjacent side — the shorter side next to the angle of interest (in this case, angle α). This side is adjacent to both the angle of interest and the right angle.
- Opposite side — simply the side opposite to the angle of interest.
- Hypotenuse — the side opposite the right angle; it's always the longest side in the right triangle.
So if our angle of interest changes to β, then the adjacent and opposite sides will be exchanged — but the hypotenuse stays the same.
The sine of an angle is the length of the opposite side divided by the length of the hypotenuse, with the assumption that the angle is acute (0° < α < 90°, or 0 < α < π/2).
sin(α) = opposite / hypotenuse = a / c
The other sine definition is based on the unit circle: the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. This definition extends sine to any angle, not just acute ones.
Important properties of the sine function:
- The range (codomain) of sine is −1 ≤ sin(α) ≤ 1;
- The sine period is equal to 2π, so sin(α + 360°) = sin(α);
- It's an odd function (while cosine is even!), which means that sin(−α) = −sin(α); and
- Sine and cosine are cofunctions: sin(α) = cos(90° − α).
Sine Curve — Sine Waves
When you plot the sine of every angle, you get the sine curve — a smooth, repeating wave called a sinusoid. It oscillates between −1 and 1 with a period of 2π. Sine waves describe an enormous range of natural phenomena: sound, light, alternating current, ocean waves, and the motion of a pendulum are all modelled with sine waves.
Sine Graph and Table (sin 0, sin 30 degrees…)
The sine function starts at 0 when the angle is 0, rises to its maximum of 1 at 90°, returns to 0 at 180°, drops to its minimum of −1 at 270°, and comes back to 0 at 360°.
The exact sine value is particularly easy to remember for certain angles — you probably learned that sin 0° = 0, sin 30° = 1/2, or sin 45° = √2/2. Other basic angles are shown in the table below.
| α (degrees) | α (radians) | sin(α) — exact | sin(α) — decimal |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 15° | π/12 | (√6 − √2) / 4 | 0.2588190451 |
| 30° | π/6 | 1/2 | 0.5 |
| 45° | π/4 | √2/2 | 0.7071067812 |
| 60° | π/3 | √3/2 | 0.8660254038 |
| 75° | 5π/12 | (√6 + √2) / 4 | 0.9659258263 |
| 90° | π/2 | 1 | 1 |
| 105° | 7π/12 | (√6 + √2) / 4 | 0.9659258263 |
| 120° | 2π/3 | √3/2 | 0.8660254038 |
| 135° | 3π/4 | √2/2 | 0.7071067812 |
| 150° | 5π/6 | 1/2 | 0.5 |
| 165° | 11π/12 | (√6 − √2) / 4 | 0.2588190451 |
| 180° | π | 0 | 0 |
Remember the periodicity of the sine function — sin(α + 360°) = sin(α) — if your angle is not shown in the table above.
Sine Calculator — How to Use
- Keep the mode on Angle → sine.
- Enter the angle in the input field. For example, type 30.
- Choose the angle unit — degrees, radians, or gradians.
- Click Calculate. The calculator instantly shows the sine value: sin(30°) = 1/2 = 0.5.
- It also shows the same angle converted to every unit, the related trig values (cos, tan, csc), and the quadrant the angle lands in.
Want to go the other way? Switch the mode to Sine value → angle, type a number between −1 and 1, and the calculator returns the angle in the <−90°, 90°> range (this is the arcsine, or inverse sine).
FAQs
What is the sine of 0, 30, 45, 60, and 90 degrees?
sin 0° = 0, sin 30° = 1/2 = 0.5, sin 45° = √2/2 ≈ 0.7071, sin 60° = √3/2 ≈ 0.8660, and sin 90° = 1. These five values are the most commonly used and worth memorizing.
Is sine an even or odd function?
Sine is an odd function, meaning sin(−α) = −sin(α). Its graph is symmetric about the origin. Cosine, by contrast, is an even function.
What is the range of the sine function?
The sine of any real angle always lies between −1 and 1 (inclusive): −1 ≤ sin(α) ≤ 1. There is no angle whose sine is greater than 1 or less than −1 — which is why the inverse mode only accepts values in that interval.
How are sine and cosine related?
They are cofunctions: sin(α) = cos(90° − α). The sine of an angle equals the cosine of its complement. Together with the Pythagorean identity sin²(α) + cos²(α) = 1, this links the two functions at every angle.
How do I convert between degrees, radians, and gradians?
A full circle is 360° = 2π rad = 400 grad. To convert degrees to radians, multiply by π/180; to convert degrees to gradians, multiply by 10/9. This calculator shows all three units automatically.