Trigonometry
Trigonometry here covers radian measure, identities, compound and double angles, the harmonic form R cos(θ ± α), and solving equations over a stated interval. Marking rewards complete solution sets, so the skill of generating every solution in range deserves practice until it is automatic.
Radians and arcs
- π radians = 180°; convert with ×π/180 or ×180/π.
- Arc length s = rθ; sector area ½r²θ; segment area = ½r²(θ − sin θ).
- Angular speed ω = θ/t, and linear speed v = rω.
Identities
- sin²θ + cos²θ = 1 and tan θ = sin θ/cos θ.
- 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.
- sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.
Harmonic form
Writing a sin θ + b cos θ = R sin(θ + α) needs R = √(a² + b²) and tan α = b/a with α acute. Confirm the arrangement by expanding R sin(θ + α) back out before committing to it.
Worked example
Express 3 sin θ + 4 cos θ in the form R sin(θ + α), then solve 3 sin θ + 4 cos θ = 2 for 0° ≤ θ < 360°. R = √(9 + 16) = 5 and tan α = 4/3, so α = 53.13°. The equation becomes 5 sin(θ + 53.13°) = 2, so sin(θ + 53.13°) = 0.4. Then θ + 53.13° = 23.58° or 156.42°, with a further 360° available. The valid solutions are θ = 103.3° and θ = 330.4°.
Traps
- Dividing both sides by cos θ and losing the solutions where cos θ = 0.
- Giving only the principal value when the interval allows more solutions.
- Leaving the calculator in the wrong mode when α appears in radians.
- Expanding cos 2θ in the wrong variant, producing an equation that will not simplify.