Trigonometry
Trigonometry tests radian and degree measure, the unit circle, identities, trigonometric equations, and the sine and cosine rules in non-right triangles.
Core rules
- Convert with π radians = 180°. Arc length s=rθ and sector area A=½r²θ require θ in radians.
- Unit-circle definitions: cos θ=x, sin θ=y, tan θ=y/x. Signs depend on the quadrant (ASTC).
- Pythagorean identity sin²θ+cos²θ=1; also tan θ=sin θ/cos θ and 1+tan²θ=sec²θ.
- Sine rule: a/sin A=b/sin B=c/sin C. Cosine rule: a²=b²+c²−2bc cos A. Area = ½ab sin C.
- General solutions repeat: sin θ=k gives θ=α+2nπ or θ=π−α+2nπ; cos θ=k gives θ=±α+2nπ.
Worked example
Solve 2sin²x−sin x−1=0 for 0≤x<2π. Factor: (2sin x+1)(sin x−1)=0. So sin x=−1/2 or sin x=1. For sin x=−1/2, reference angle π/6, solutions in QIII and QIV: x=7π/6, 11π/6. For sin x=1, x=π/2. Solution set: {π/2, 7π/6, 11π/6}.
Traps
Mixing degrees and radians is a common error; match the mode or convert first. Do not discard valid solutions when taking a square root; include both signs. Check whether the required interval is closed or open. In the sine rule, the ambiguous case (two possible triangles) can arise when given two sides and a non-included angle. Never divide by sin x or cos x without checking the zero case.