Pure: trigonometry

3.1 · Free notes

Radians, trigonometry and parametric equations

Key points

• Radians: 180° = π radians. Arc length s = rθ; area of a sector = ½r²θ, with θ in radians.
• Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
• Double angle formulae: sin2θ = 2sinθcosθ; cos2θ = cos²θ − sin²θ; and the compound angle formulae for sin(A ± B) and cos(A ± B).
• The form R sin(θ ± α) or R cos(θ ± α) is used to solve equations like a sinθ + b cosθ = c and to find maxima and minima.
• Parametric equations define x and y in terms of a parameter t; converting to Cartesian form means eliminating t.

In more detail

• Worked identity equation: sec²θ = 3 tan θ − 1 → tan²θ − 3 tan θ + 2 = 0 → tan θ = 1 or 2 → θ = 45°, 63.4°, 225°, 243.4°.
• Worked double angle: sin 2θ = sin θ → sin θ(2 cos θ − 1) = 0 → θ = 0°, 60°, 180°, 300°.
• Worked R form: 3 sin x + 4 cos x = 5 sin(x + 53.1°), so its maximum is 5, when x = 36.9°.
• Worked parametric: x = 2t, y = t² → y = x²/4; x = 3 cos t, y = 3 sin t → x² + y² = 9.
• Worked sector: an arc of 5 cm with r = 4 cm → θ = 1.25 rad and area = ½ × 16 × 1.25 = 10 cm².