Trigonometry

• sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ.
• sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ.
• Compound angles: sin(A ± B), cos(A ± B), tan(A ± B) — in the formula list, but you must use them fluently.
• Double angles: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2tan A/(1 − tan²A).
• a sin θ + b cos θ can be written as R sin(θ ± α) or R cos(θ ± α), with R = √(a² + b²).

Graphs, identities and equations

The graph of y = sec θ has vertical asymptotes wherever cos θ = 0 (θ = 90°, 270°, …), and |sec θ| ≥ 1 everywhere. y = cosec θ has asymptotes where sin θ = 0 (0°, 180°, …), also with |cosec θ| ≥ 1. y = cot θ has asymptotes where tan θ = 0 and crosses the axis where tan θ has its asymptotes.

Divide sin²θ + cos²θ ≡ 1 by cos²θ or by sin²θ to get the two new Pythagorean identities. Use them to turn an equation into a quadratic in one function.

Expanding, collapsing and the R form

Compound-angle formulae give exact values such as sin 15° or tan 75°, and turn expressions like cos(x − 30°) into terms in sin x and cos x. Double-angle formulae replace a 2θ with θ; choose the version of cos 2A that leaves only one function in the equation.

To write a sin θ + b cos θ as a single wave, expand the target form and compare coefficients. Then the maximum and minimum are ±R, and an equation a sin θ + b cos θ = c becomes one equation in (θ ± α).