Trigonometric Functions and Trigonometric Ratios
What the exam tests
Trigonometric Functions and Trigonometric Ratios tests radian measure, the sine and cosine graphs, identities, general solutions, and triangle ratios. Kyoto questions often combine an equation with a restricted interval or ask for exact values without a calculator.
Core rules
- In radians, sin and cos have period 2π; tan has period π. Memorize exact values at 0, π/6, π/4, π/3, π/2 and use reference angles.
- Pythagorean identity: sin^2 x + cos^2 x = 1. Also tan x = sin x / cos x when cos x ≠ 0.
- For sin x = s with -1 ≤ s ≤ 1, solutions in one period must be found in both relevant quadrants; do not divide by sin x or cos x without checking the zero case.
- In a triangle, the law of sines is a/sin A = b/sin B = c/sin C, and the law of cosines is c^2 = a^2 + b^2 - 2ab cos C.
Worked example
Solve 2sin^2 x - sin x - 1 = 0 for 0 ≤ x < 2π. Factor as (2sin x + 1)(sin x - 1) = 0. Thus sin x = -1/2 or sin x = 1. For sin x = -1/2, the solutions are x = 7π/6 and 11π/6. For sin x = 1, the solution is x = π/2. The solution set is {π/2, 7π/6, 11π/6}.
Traps
Mixing degrees and radians is a common error. When taking an inverse sine, remember the second solution in the interval. Check whether tan is undefined at a candidate. Do not replace sin^2 x by sin x^2.