Trigonometry
Trigonometry in SEE Optional Mathematics tests identity manipulation, solution of trigonometric equations within a stated interval, and modelling heights and distances with right triangles.
Identities
Start from sin²θ+cos²θ=1; divide by cos²θ to get 1+tan²θ=sec²θ, by sin²θ to get 1+cot²θ=csc²θ. For compound angles, use sin(A±B)=sinA cosB ± cosA sinB and cos(A±B)=cosA cosB ∓ sinA sinB. When proving an identity, work on one side only until it matches the other; convert everything to sine and cosine if stuck.
Equations
Solve within the given range, usually 0°≤θ<360° or 0≤θ<2π. Find the reference angle, then use the ASTC sign pattern (All, Sine, Tan, Cos positive in quadrants I–IV) to list all solutions. For equations like sinθ=k, θ=α+360°n or 180°−α+360°n.
Heights and distances
Draw a clear diagram. Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal. In a right triangle, tanθ=opposite/adjacent, sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse. For two observers or two buildings, introduce a common height and set up simultaneous equations.
Worked example
From a point 30 m from the foot of a vertical pole, the angle of elevation of the top is 45°. Height h satisfies tan45°=h/30, so h=30 m.
Traps
Using the depression angle inside the triangle without converting it to the elevation angle at the object; forgetting the second solution in 0°–360°; rounding too early.