4-10 Finding an angle from a given ratio

Acute angle first

• Ignore the sign, find the acute reference angle from the special values or the tables, then restore the quadrant.

All solutions on 0° ≤ θ < 360°

• Fix the two quadrants where the function has the given sign, then write θ = reference angle and θ = 180° ± reference or 360° − reference as appropriate.
• Example: cosθ = −½ with 0° ≤ θ < 360°: reference 60°, cosine negative in the second and third quadrants → θ = 120° or θ = 240°.
• In radians the same idea gives, for sinθ = ½ on 0 ≤ θ < 2π: θ = π/6 or θ = 5π/6.

Simple equations

• Isolate the function first: 2sinθ − 1 = 0 → sinθ = ½; never divide the whole equation by the function name.
• For quadratic-type equations such as 2cos²θ − cosθ − 1 = 0, put u = cosθ, factor (2u + 1)(u − 1) = 0 and solve each branch with its own quadrants.
• Reject any value outside the stated interval, and check that |sin|, |cos| ≤ 1 — a ratio like sinθ = 1.5 has no solution.