Trigonometry
What the domain tests
Trigonometry tests identities, trigonometric equations, solution of triangles and radian measure. MSCE Additional Mathematics commonly combines a Pythagorean identity with a quadratic equation, or asks for every solution in a stated interval, so both algebraic and graphical thinking are needed.
Core rules
- sin^2θ+cos^2θ=1 and tanθ=sinθ/cosθ. Compound angles: sin(A±B)=sinAcosB±cosAsinB; cos(A±B)=cosAcosB∓sinAsinB.
- Sine rule: a/sinA=b/sinB=c/sinC. Cosine rule: a^2=b^2+c^2-2bc cosA. Area=1/2 ab sinC.
- In radians, arc length s=rθ and sector area A=1/2 r^2θ, with θ in radians.
- For equations, reduce to one trigonometric function, factorise, then use quadrant signs or a cast diagram to list all angles in the required interval.
Worked example
Solve 2sin^2x+sinx-1=0 for 0°≤x<360°. Let u=sinx. Then 2u^2+u-1=0=(2u-1)(u+1), so sinx=1/2 or sinx=-1. For sinx=1/2, x=30° or 150°. For sinx=-1, x=270°. Solutions: 30°,150°,270°.
Traps
Using degree mode when the interval is in radians; dividing by sinx and losing solutions; forgetting the second-quadrant solution; applying the cosine rule with the wrong included angle.