Sine Calculator
Solve sine ratio or angle.
About the Sine Calculator
Sine relates an angle to the ratio of the opposite side over the hypotenuse in a right triangle, and describes every wave from sound to alternating current.
How to use it
- Enter the angle.
- Note that the input is in radians.
- Read the sine value.
The formula
In a right triangle, sine is the ratio of the side opposite an angle to the hypotenuse. The mnemonic is SOH — Sine, Opposite, Hypotenuse.
sin(θ) = Opposite ÷ Hypotenuse
θ = arcsin(Opposite ÷ Hypotenuse)
Because the hypotenuse is always the longest side, sine can never exceed 1. If you get a ratio above 1, you have divided by the wrong side.
The deeper definition is the unit circle. Take a circle of radius 1 and rotate a point around it by angle θ: the sine is the point's height. That is why sine keeps working past 90°, where no right triangle exists, and why it produces a wave — the height rises to 1, falls through 0 to −1, and repeats every 360°. Every oscillation in physics, from a pendulum to alternating current to sound, is described with this function.
Worked examples
| Angle | sin(θ) | Opposite | Hypotenuse | Where it shows up |
|---|---|---|---|---|
| 30° | 0.5000 | 5 | 10 | the exact half — worth memorising |
| 45° | 0.7071 | 7.071 | 10 | equal sides, √2 ÷ 2 |
| 60° | 0.8660 | 8.660 | 10 | √3 ÷ 2 |
| 90° | 1.0000 | 10 | 10 | opposite side IS the hypotenuse |
Notice the ratios do not climb evenly. Between 0° and 30° sine gains 0.5; between 60° and 90° it gains only 0.134. Sine flattens as it approaches its maximum, which is why a ramp gets much steeper for each extra unit of height near the top, and why solar panels lose surprisingly little output from small tilt errors near their optimum.
Common mistakes
- Having the calculator in radians. This is the single most common trigonometry error. sin(30) is 0.5 in degrees and −0.988 in radians. If your answer is wildly wrong and you cannot see why, check the mode first.
- Mixing up opposite and adjacent. Opposite is across the triangle from the angle you are working with. Adjacent touches it. Which side is which changes depending on the angle you have chosen, not on how the triangle is drawn.
- Trying to use SOH CAH TOA on a triangle with no right angle. These ratios are defined for right triangles only. For any other triangle you need the sine rule or the cosine rule.
- Entering a ratio greater than 1 into arcsine. It has no solution, because the opposite side cannot be longer than the hypotenuse. That error almost always means the hypotenuse and a leg were swapped.
Terms explained
- Hypotenuse
- The side opposite the right angle, and always the longest. The denominator in both sine and cosine.
- Opposite side
- The side across from the angle in question. The numerator for sine.
- Unit circle
- A circle of radius 1 centred on the origin. Sine is the height of a point rotated around it, which extends the definition to any angle.
- Arcsine
- The inverse function, written sin−¹ or asin. It takes a ratio and returns the angle.
- Radian
- The other angle unit: 2π radians make a full turn, so 180° is π. Standard in calculus and in most programming languages.
- Sine rule
- For any triangle, a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C). How you solve triangles that have no right angle.
Common questions
- Why is my answer wrong by a lot?
- Almost certainly degrees versus radians. sin(30°) is 0.5; sin(30 radians) is −0.988. Check the mode before checking anything else.
- What does SOH CAH TOA mean?
- Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. Three ratios, one mnemonic.
- Can sine be more than 1?
- Never, for a real angle. The opposite side cannot exceed the hypotenuse, so the ratio is capped at 1 — reached exactly at 90°.
- How do I find an angle from two sides?
- Divide opposite by hypotenuse, then take the arcsine. Opposite 5 and hypotenuse 10 gives 0.5, and arcsin(0.5) is 30°.
- Why does sine make a wave?
- Because on the unit circle it measures height as a point rotates. Height rises to 1, falls to −1 and repeats every full turn, so plotting it against angle traces a wave. That is why it models sound, light, tides and alternating current.
- What is the sine rule for?
- Triangles without a right angle. It relates each side to the sine of its opposite angle, which lets you solve a triangle from two angles and a side, or two sides and a non-included angle.
- How do I convert degrees to radians?
- Multiply by π and divide by 180. So 30° is about 0.524 radians. Most programming languages expect radians, which is why code that looks correct often is not.
- What are the values I should memorise?
- sin(0°) = 0, sin(30°) = 0.5, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1. They recur constantly and make it easy to sanity-check a calculator.