Trigonometry basics

Right-angle trigonometry

Pick any right triangle. Label the angle you care about θ. The side opposite θ is the opposite, the side next to it (excluding the hypotenuse) is the adjacent, and the longest side is the hypotenuse. Three ratios pop out, and they have famous names:

• sine = opposite / hypotenuse
• cosine = adjacent / hypotenuse
• tangent = opposite / adjacent

The mnemonic SOH-CAH-TOA captures all three at once. Whatever the size of the triangle, as long as the angle θ is the same, the ratios are the same. That is why trigonometry can talk about "sine of 30°" without you needing to specify the size of the triangle.

The unit circle

Take a circle of radius 1 centred at the origin. Pick a point on the circle and draw a line to it from the centre. The angle that line makes with the positive x-axis is θ. Then the x-coordinate of the point is cos θ and the y-coordinate is sin θ. That single picture lets you extend trigonometry beyond right triangles to any angle, including angles bigger than 90° and negative angles.

Three identities fall out of the unit-circle picture immediately:

• sin²θ + cos²θ = 1 — the Pythagorean identity, which is just Pythagoras applied to the unit circle.
• sin(θ) = sin(180° − θ) — reflection symmetry.
• cos(θ) = cos(−θ) — cosine is even.

Radians vs degrees

Degrees split a full circle into 360 equal slices, which is a Babylonian convention. Radians use the radius of the circle as the unit: an angle of one radian sweeps out an arc of length 1 on a unit circle. A full circle is 2π radians ≈ 6.283. School problems usually use degrees; calculus and physics almost always use radians, because the derivative of sin x is cos x only when x is in radians.

Solving any triangle

Two formulas extend trigonometry to triangles that do not have a right angle:

• The law of sines: a / sin A = b / sin B = c / sin C. Useful when you know two angles and a side, or two sides and a non-included angle.
• The law of cosines: c² = a² + b² − 2ab cos C. A generalisation of Pythagoras; useful when you know two sides and the included angle, or all three sides.

The trigonometry calculator applies these automatically based on which fields you fill in.

Two identities you will reuse forever

The angle-addition formulas come up in nearly every advanced derivation:

• sin(A + B) = sin A cos B + cos A sin B
• cos(A + B) = cos A cos B − sin A sin B

Setting A = B gives the double-angle formulas sin(2A) = 2 sin A cos A and cos(2A) = cos²A − sin²A, both of which let you simplify expressions that would otherwise need a calculator.