Sine Calculator

What this calculator does

So, you have an angle and you want its sine. This tool takes the angle, in degrees or radians, and returns the sine value rounded to three decimal places.

There is one input box for the angle and a dropdown to say whether that angle is in degrees or radians. Everything else is a single button press.

How to use it

1. Type your angle.
2. Choose degrees or radians from the dropdown to match it.
3. Press Calculate.

Getting the unit right matters, since 90 degrees and 90 radians are completely different angles. If you picked degrees, the tool converts to radians internally before computing, because that is the unit the underlying maths works in.

What sine actually is

Sine is, at heart, a height. In a right triangle, the sine of an angle is the side opposite that angle divided by the hypotenuse, the long side. That is the first half of the old mnemonic SOH: Sine equals Opposite over Hypotenuse. So if you know an angle and the hypotenuse, the sine tells you how tall the opposite side is as a fraction of it.

Because it is a ratio of two lengths, sine has no units of its own. It is just a number, and for any given angle that number is fixed, no matter how big or small the triangle is.

Sine on the unit circle

The triangle picture only reaches to 90 degrees, so for every angle, sine is defined on the unit circle, a circle of radius 1 centred at the origin. Take the angle, measured from the positive x-axis, and find the point where it lands on the circle. The sine of the angle is simply the height of that point, its y-coordinate. This is why sine runs smoothly through all angles, past 90 degrees and beyond a full turn: the point just keeps going around, and its height keeps rising and falling.

Its range, and the wave it draws

Since the point never leaves a circle of radius 1, its height can never exceed 1 or drop below -1. So the sine of any angle always sits between -1 and 1. It reaches 1 at 90 degrees, the top of the circle, falls to 0 at 180 degrees, and bottoms out at -1 at 270 degrees. Plot sine against the angle and you get the smooth, rolling sine wave, the shape that describes almost everything that oscillates: sound, light, tides, a swinging pendulum, the alternating current in the wall socket. That is why sine matters far beyond triangles.

A worked example

Take 30 degrees. On the unit circle, the point at 30 degrees sits at a height of exactly one half, so the sine of 30 degrees is 0.5, which the tool shows as 0.500. At 90 degrees the point is at the very top, height 1, so the sine is 1. At 0 degrees the point is on the axis, no height at all, so the sine is 0.

Degrees or radians

The dropdown matters because the two units count angles differently. Degrees split a full turn into 360 parts, the everyday way. Radians measure the angle by arc length around the unit circle, so a full turn is 2 pi radians, and a right angle is pi over 2. Higher mathematics almost always uses radians, because the formulas come out cleaner, while degrees are more familiar day to day. Pick whichever your angle is written in, and the tool handles the rest.

Questions people ask

What does the sine of an angle tell me?

In a right triangle, it is the opposite side divided by the hypotenuse. On the unit circle, it is the height of the point at that angle. Either way it is a single number that depends only on the angle.

Why is sine always between -1 and 1?

Because it is the height of a point on a circle of radius 1, and that height cannot go above 1 or below -1.

Does sine have units?

No. It is a ratio of two lengths, so the units cancel and the answer is a plain number.

Degrees or radians, which should I choose?

Whichever your angle is given in. Degrees for everyday angles, radians for most higher maths. Just match the dropdown to your value, since 90 degrees and 90 radians are very different.

How is sine related to cosine?

They are partners: sine is the height of the unit-circle point and cosine is its horizontal position. The cosine calculator handles that companion function.

References

On the sine and where its name comes from. Sine is the vertical coordinate of a point on the unit circle, and the word itself has a winding history, traced from the Sanskrit jya through Arabic to a Latin mistranslation.

1. Eric W. Weisstein, "Sine," from MathWorld, a Wolfram resource, on the sine as the vertical coordinate of an arc endpoint on the unit circle.
2. "Trigonometric functions," MacTutor History of Mathematics, on Aryabhata's early sine tables and the path from jya to the Latin sinus.