Pythagorean identity: sin²θ + cos²θ = 1 for every angle θ. Quotient identity: tan θ = sin θcos θ (when cos θ ≠ 0). Rearranged forms are useful: sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ.

When you take a square root, use the quadrant of θ to choose the sign. Sine is positive in quadrants 1 and 2, cosine in quadrants 1 and 4, and tangent in quadrants 1 and 3.

Worked example. sin θ = 35 with θ in the second quadrant. cos²θ = 1 − 925 = 1625, and cosine is negative there, so cos θ = −45 and tan θ = 35 ÷ (−45) = −[frac 3/4].

Method: make the trigonometric ratio the subject, find the reference angle (the acute angle with that ratio, ignoring the sign), then use the signs in each quadrant to place every solution in the domain.

For an equation in 2x (or 3x, …), first change the domain: if 0 ≤ x ≤ 2π then 0 ≤ 2x ≤ 4π. Find every value of 2x in that domain, then divide by 2.

Worked example. Solve tan x = −1 for 0 ≤ x ≤ 2π. The reference angle is π4, and tangent is negative in quadrants 2 and 4, so x = π − π4 = [frac 3π/4] or x = 2π − π4 = [frac 7π/4].