Trigonometry
The Trigonometry domain tests the unit circle, identities, graphs, and triangle-solving techniques.
Unit circle and identities
On the unit circle the coordinates are (cos θ, sin θ), so sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ. Dividing that identity by cos²θ or sin²θ yields 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. Sum, difference, and double-angle formulas handle combined arguments.
Graphs
For y = A sin(B(x − C)) + D, the amplitude is |A|, the period is 2π/|B|, the phase shift is C, and the midline is y = D.
Solving triangles
Law of Sines: a/sin A = b/sin B = c/sin C, with an ambiguous SSA case that can produce 0, 1, or 2 triangles. Law of Cosines: c² = a² + b² − 2ab cos C.
Worked example
Solve sin(2x) = 1/2 for 0 ≤ x < 2π. Reference solutions are 2x = π/6 + 2kπ and 2x = 5π/6 + 2kπ. Dividing by 2 gives x = π/12 + kπ and x = 5π/12 + kπ. Inside the interval these are x = π/12, 5π/12, 13π/12, 17π/12 - four solutions, not two, because halving the argument halved the period.
Traps
Reporting only the principal solution and missing others in the interval; forgetting the ambiguous SSA case; leaving the calculator in the wrong angle mode; confusing a phase shift inside the parentheses with a vertical shift outside.