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Trigonometry formulas

Trigonometry connects angles to lengths. It starts with the ratios in a right triangle, extends to any triangle through the laws of sines and cosines, and ends with sine waves that describe anything that repeats. Every calculator here works in degrees or radians: use the toggle above the inputs.

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SOH CAH TOA (right-triangle trigonometry)

sin⁡θ=oppositehypotenuse, cos⁡θ=adjacenthypotenuse, tan⁡θ=oppositeadjacent

The three trigonometric ratios of an angle in a right triangle; give any two of angle and sides and they find the rest.

Pythagorean identity

sin2⁡θ+cos2⁡θ=1

For every angle, the squares of its sine and cosine add up to 1, so one gives the other up to a sign.

Double-angle formulas

sin⁡2θ=2sin⁡θcos⁡θ, cos⁡2θ=cos2⁡θ−sin2⁡θ

The sine and cosine of twice an angle, written using the sine and cosine of the angle itself.

Angle sum and difference formulas

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B, cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B

The sine and cosine of a sum or difference of two angles, in terms of the sines and cosines of each.

Law of sines

asin⁡A=bsin⁡B=csin⁡C

In any triangle, each side divided by the sine of the opposite angle gives the same number.

Law of cosines

c2=a2+b2−2abcos⁡C

Pythagoras for any triangle: the third side from two sides and the angle between them, or any angle from three sides.

Area of a triangle from two sides and an angle

A=12absin⁡C

The area of any triangle from two sides and the angle between them, no height needed.

Degrees and radians

radians=degrees×π180

Converts angles between degrees and radians: 180° is π radians.

Sine wave (amplitude, period and phase)

y=Asin⁡(B(x−C))+D

The general sine wave: A stretches it vertically, B sets how often it repeats, C slides it sideways and D lifts it.