Trigonometry
Trigonometry covers trigonometric ratios, identities, graphs, and equations and inequalities. Course 2 questions often mix radian measure with identities and periodic solutions.
Core rules
- sin^2θ+cos^2θ=1; tanθ=sinθ/cosθ. Double-angle forms include sin2θ=2sinθcosθ and cos2θ=1-2sin^2θ.
- Radians: π=180°. The sine and cosine functions have period 2π; tan has period π.
- For equations, isolate one trigonometric function or factor into products, then list solutions in the required interval.
- Graphs help check the number of solutions and whether endpoints are included.
Worked example
Solve 2sin^2x-sinx-1=0 for 0≤x<2π. Let t=sinx: 2t^2-t-1=(2t+1)(t-1)=0, so t=1 or t=-1/2. In the interval, sinx=1 gives x=π/2; sinx=-1/2 gives x=7π/6 and x=11π/6. The solution set is {π/2,7π/6,11π/6}.
Traps
Do not drop the second solution from a periodic function. Do not square both sides unless you check for extraneous roots. When the interval is in degrees, answer in degrees; when in radians, answer in radians. Remember that tan is undefined where cos=0, and that cos2x has period π even though cos x has period 2π.