Trigonometry connects angles to lengths. These guides cover the unit circle and exact values, the identities you are expected to know, and the laws used to solve any triangle.
Right-triangle trigonometry, with sine as opposite over hypotenuse, only works for angles between 0° and 90°. The unit circle extends the same functions to every angle, and the identities and triangle laws follow from it. The guides here are arranged in that order.
Every article gives exact values where they exist, explains when to work in degrees and when in radians, and includes fully worked triangle problems with answers checked by a second method. The ambiguous SSA case of the law of sines gets its own careful treatment because it is where most exam marks are lost.
Angles in these articles are given in both degrees and radians wherever it helps, and every calculator result is quoted with the mode it was computed in, since a calculator left in the wrong mode is the most common source of impossible trigonometry answers.
Begin with the unit circle, then identities, then the law of sines and the law of cosines, and finish with the general guide to solving triangles.
Trigonometry
The unit circle turns trigonometry from a list of facts to memorise into a picture you can read. Every angle corresponds to a point on the circle, and that point's coordinates are the cosine and sine of the angle.
Trigonometry
A trigonometric identity is an equation that holds for every angle where both sides are defined. A short list of identities, all built from the unit circle, lets you simplify expressions, find exact values, prove results and solve equations.
Trigonometry
The law of sines says that in any triangle, each side divided by the sine of its opposite angle gives the same number. It solves triangles when you know two angles and a side, and, with care, when you know two sides and an angle opposite one of them.
Trigonometry
The law of cosines extends the Pythagorean theorem to every triangle. It finds the third side when you know two sides and the angle between them, and it finds any angle when you know all three sides.
Trigonometry
To solve a triangle is to find all three sides and all three angles from the parts you are given. Which tool to use depends on which three parts you know, so the first step is always to identify the case.