Trigonometry Calculator

What this calculator does

This is two trig tools in one. A right-triangle solver, where you enter any two values of a right triangle and it finds the rest. And a trig-function evaluator, where you enter an angle and it gives all six trigonometric functions.

Use whichever half you need, or both.

Using the calculator

1. Triangle solver: enter any two of Side A, Side B, Hypotenuse, Angle A, or Angle B. Angle C is the 90° right angle. Press Calculate and the remaining sides and angles fill in.
2. Function evaluator: enter an angle in degrees, press Calculate, and you get sine, cosine, tangent, and their reciprocals.

The triangle solver needs exactly two known values to work from.

The three ratios: SOH-CAH-TOA

The engine behind all of this is the three trigonometric ratios, which link an angle to a ratio of the triangle's sides. They are remembered by the word SOH-CAH-TOA:

• Sine = Opposite ÷ Hypotenuse
• Cosine = Adjacent ÷ Hypotenuse
• Tangent = Opposite ÷ Adjacent

Opposite and adjacent are named relative to the angle you are looking at, while the hypotenuse is always the longest side, sitting opposite the right angle.

Solving a right triangle

With any two values known, these ratios together with the Pythagorean theorem, side A squared plus side B squared equals the hypotenuse squared, pin down everything else. Know two sides? Pythagoras gives the third, and the ratios give the angles. Know a side and an angle? The ratios give the other sides. That is exactly what the solver does behind the scenes.

All six functions of an angle

The function half gives all six trigonometric values of an angle at once. The three main ones, sine, cosine, and tangent, plus their three reciprocals: cosecant (1 ÷ sine), secant (1 ÷ cosine), and cotangent (1 ÷ tangent). It is worth noting that the tangent is itself the sine divided by the cosine, which ties the trio together.

Two worked examples

Triangle: with Side A = 3 and Side B = 4, Pythagoras gives the hypotenuse as √(9 + 16) = 5, and the two acute angles come out to about 36.87° and 53.13°. This is the classic 3-4-5 triangle.

Functions: at 30°, the sine is about 0.5, the cosine about 0.87, and the tangent about 0.58.

Going deeper on one function

For a single function on its own, there are focused tools: the sine, cosine, and tangent calculators. For the reciprocals, see the secant and cotangent calculators.

Questions people ask

What does this calculator do?

It solves a right triangle from any two known values, and it evaluates all six trigonometric functions of an angle.

What is SOH-CAH-TOA?

A way to remember the three ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

How many values do I need for the triangle?

Two. From any two of the sides and angles, the ratios and Pythagoras give the rest.

Does it give all six functions?

Yes: sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent.

References

A note on the idea behind it. The trigonometric ratios sine, cosine, and tangent relate an angle to ratios of a right triangle's sides, summarised by SOH-CAH-TOA. With two known values, these ratios and the Pythagorean theorem determine the remaining sides and angles. For further reading, see Trigonometry.

1. The trigonometric ratios, sine, cosine, and tangent as ratios of a right triangle's sides.
2. Solving a right triangle, using the ratios and the Pythagorean theorem from two known values.