Sine Calculator

What the sine function measures

The sine of an angle is a number between −1 and 1 that describes how high the angle's far end sits above the horizontal. In a right triangle with an acute angle θ, it is the length of the side opposite θ divided by the length of the hypotenuse.

sin θ = opposite ÷ hypotenuseValid for the acute angles of a right triangle; the unit circle extends it to every angle.

On the unit circle (radius 1, centred at the origin), turn counter-clockwise from the positive x-axis by θ and you land on the point (cos θ, sin θ). The sine is simply the height of that point, which is why it rises from 0 to 1 over the first quarter turn, falls back to 0 at a half turn, dips to −1 at three quarters and returns to 0 after a full turn.

• Domain: every real angle, in degrees or radians.
• Range: from −1 to 1, both included.
• Period: 360° (2π radians), so sin(θ + 360°) = sin θ.
• Symmetry: sine is an odd function, so sin(−θ) = −sin θ.

How to use this calculator

1. Type the angle into the Angle box. Negative angles and decimals are fine.
2. Choose degrees or radians in the list next to it. Degrees is the default.
3. Press Calculate. The answer appears as sin(angle) = value.

The result is rounded to six decimal places. Values that come out as whole numbers are written without decimals, so sin(90°) shows 1 and sin(180°) shows 0, even though a computer's raw arithmetic lands a hair away from those figures. A value with only one or two decimals is padded to two (sin 30° = 0.50), one with three or four decimals is padded to four (0.1250), and anything longer keeps six, as in sin 45° = 0.707107.

The angle must be written as a plain number such as 30, -12.5 or 0.0001. Exponent notation (1e3), a leading plus sign, a bare leading or trailing decimal point and thousands separators are not accepted and trigger the error message instead of a result.

Sine of common angles

A handful of angles have sines you can write exactly. Those for 30°, 45° and 60° follow a pattern worth remembering: they are √1 ÷ 2, √2 ÷ 2 and √3 ÷ 2.

Finding a sine without a calculator

For angles that are multiples of 30° or 45°, reduce the angle first and read the answer from the table above. Add or subtract multiples of 360° until the angle lies between 0° and 360°, find its reference angle (the acute angle it makes with the x-axis) and attach the sign: sine is positive between 0° and 180° and negative between 180° and 360°.

Example: sin 750°. Subtract 720° to get 30°, and sin 30° = 0.5. Likewise sin(−210°) is the same as sin 150°, whose reference angle is 30°, so it is also 0.5.

For any other angle, convert to radians (multiply degrees by π ÷ 180) and sum the series below. Each extra term corrects the previous estimate by a smaller amount, so a few terms are enough for four or five digits when the angle is small.

sin x = x − x³/3! + x⁵/5! − x⁷/7! + …x must be in radians.

Example: x = 0.5 radian. The terms are 0.5, −0.125 ÷ 6 = −0.020833, +0.03125 ÷ 120 = +0.000260 and −0.0078125 ÷ 5040 = −0.000002. Adding them gives 0.479426, which matches the calculator's 0.479426.

Worked example: how high does a ladder reach?

A 5 m ladder leans against a wall and makes a 70° angle with the level ground. The wall is vertical, so the height reached is the side opposite the 70° angle and the ladder is the hypotenuse.

height = ladder length × sin 70° = 5 × 0.939693 = 4.698 m

Enter 70 with degrees selected to get 0.939693, then multiply by 5. Had the unit been left on radians, the calculator would have answered sin(70) = 0.773891 and the ladder would seem to reach only 3.87 m, which is why the unit selector deserves a glance every time.

Useful identities and related functions

• Pythagorean identity: sin²θ + cos²θ = 1, so the cosine of an angle can be recovered from its sine up to a sign.
• Complementary angles: sin θ = cos(90° − θ).
• Double angle: sin 2θ = 2 sin θ cos θ.
• Law of sines: in any triangle, a ÷ sin A = b ÷ sin B = c ÷ sin C. If a = 10 with A = 40° and B = 65°, then b = 10 × sin 65° ÷ sin 40° ≈ 14.10.
• Inverse: to go from a sine value back to an angle, use the arcsine calculator.

The ratios of sine and cosine give the tangent and cotangent, and the trigonometry calculator solves a whole triangle from the parts you know. More tools are collected under trigonometry.

Where sine shows up

• Heights and distances in surveying, construction and navigation, wherever a slanted length and an angle are known.
• Waves of every kind: sound, light, alternating current and vibrations are modelled as sine curves, y = A sin(ωt + φ).
• Physics: the component of a force along a slope is F sin θ, and the vertical speed of a projectile launched at angle θ is v sin θ.
• Computer graphics and games, where sine drives rotation, oscillation and smooth animation.