Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — an exact value from a compound angle
Find the exact value of sin75°.
- 1Write 75° as a sum of angles whose values you know: 75=45+30.60+15 would not help — 15° is not a standard angle.
- 2sin(45°+30°)=sin45°cos30°+cos45°sin30°The sine formula keeps the sign given.
- 3=21⋅23+21⋅21
- 4=223+1Common denominator.
- 5Rationalise: =46+2Multiply top and bottom by 2.
- 6Check: 42.449+1.414=0.966=sin75° ✓A numerical check catches sign errors instantly.
Example 2 — the R-formula
Express 3cosθ+4sinθ in the form Rcos(θ−α), and hence solve 3cosθ+4sinθ=2 for 0°≤θ<360°.
- 1R=a2+b2=9+16=5Always positive.
- 2tanα=ab=34, so α=53.13°Sine coefficient over cosine coefficient.
- 33cosθ+4sinθ=5cos(θ−53.13°)The maximum is 5 and the minimum −5.
- 45cos(θ−53.13°)=2, so cos(θ−53.13°)=0.4
- 5θ−53.13°=±66.42°Cosine is positive in the first and fourth quadrants.
- 6θ=119.6° or θ=−13.3°≡346.7°Add 360° to bring the second into range. Both check to 2 ✓
Example 3 — a double angle equation
Solve sin2θ=sinθ for 0≤θ<2π.
- 1Replace sin2θ with 2sinθcosθ.Reduce to a single angle first.
- 22sinθcosθ−sinθ=0Move everything to one side.
- 3sinθ(2cosθ−1)=0Factorise — never divide by sinθ, or two solutions vanish.
- 4sinθ=0 gives θ=0 and θ=π.
- 5cosθ=21 gives θ=3π and 35π.Cosine is positive in quadrants 1 and 4.
- 6Four solutions: 0,3π,π,35π