Trigonometry
What this domain tests
Trigonometric functions, identities, equations and triangle geometry. The 5-period exam expects radian measure, exact values, transformation of expressions, and complete solution sets over a stated interval.
Core rules
- Radians: pi radians = 180 degrees. Arc length s = r(theta) and sector area A = (1/2)r^2(theta) for theta in radians.
- Exact values: sin(pi/6) = 1/2, cos(pi/3) = 1/2, sin(pi/4) = cos(pi/4) = sqrt(2)/2, sin(pi/3) = sqrt(3)/2, tan(pi/4) = 1.
- Pythagorean identity: sin^2 x + cos^2 x = 1; tan x = sin x/cos x.
- Double angle: sin 2x = 2 sin x cos x; cos 2x = cos^2 x - sin^2 x = 1 - 2 sin^2 x = 2 cos^2 x - 1.
- Graphs: y = a sin(bx + c) + d has amplitude |a|, period 2pi/|b|, phase shift -c/b and vertical shift d.
- Triangle rules: sine rule a/sin A = b/sin B = c/sin C; cosine rule a^2 = b^2 + c^2 - 2bc cos A; area = (1/2)ab sin C.
Worked example
Solve 2 sin^2 x - sin x - 1 = 0 for 0 <= x < 2pi. Let t = sin x. Then 2t^2 - t - 1 = (2t + 1)(t - 1) = 0, so t = -1/2 or t = 1. Now sin x = 1 gives x = pi/2. Also sin x = -1/2 gives x = 7pi/6 and x = 11pi/6, because sine is negative in the third and fourth quadrants and the reference angle is pi/6. The solution set is {pi/2, 7pi/6, 11pi/6}.
Traps that cost marks
- Working in degrees when the question specifies radians, or vice versa.
- Dividing an equation by sin x or cos x and losing the solutions where that factor is zero.
- Reporting only the principal-value solution instead of all solutions in the interval.
- Using the wrong sign for cos 2x, or forgetting that inverse sine has range from -pi/2 to pi/2.