Trigonometric identities and equations

Learning objectives

State and use the identities tan x = sin x / cos x, sin²x + cos²x = 1, sec²x = 1 + tan²x and cosec²x = 1 + cot²x, and derive the last two from the first Pythagorean identity

Prove a trigonometric identity by working on one side only, converting everything into sines and cosines or using a Pythagorean identity, until it matches the other side

Solve a trigonometric equation within a stated interval by finding the principal value and then using the symmetry and period of the graph to obtain every solution in range

Solve equations that reduce to a quadratic in a single trigonometric ratio, factorising and rejecting any value outside the range of that ratio, such as sin x = 1.5

Solve equations of the form a sin(bx + c) = k by adjusting the interval for the whole argument before solving, and converting back at the end

Work in radians throughout when the interval is given in radians, and give answers to the accuracy demanded by the question