Pure 1: Trigonometry

Pure 1: Trigonometry — CIE A Level Mathematics 9709

1. Radians and Arc/Sector Formulae

Radian measure: 2π rad = 360°. Key conversions:

• π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°

Arc length: s = rθ (θ in radians)

Sector area: A = ½r²θ

Example: A sector has radius 6 cm and angle 0.8 rad. Find arc length and area.

s = 6 × 0.8 = 4.8 cm; A = ½ × 36 × 0.8 = 14.4 cm²

Segment area = Sector area − Triangle area = ½r²θ − ½r²sinθ = ½r²(θ − sinθ)

2. Exact Values

| θ | sin θ | cos θ | tan θ |

|---|---|---|---|

| 0° (0) | 0 | 1 | 0 |

| 30° (π/6) | 1/2 | √3/2 | 1/√3 |

| 45° (π/4) | 1/√2 | 1/√2 | 1 |

| 60° (π/3) | √3/2 | 1/2 | √3 |

| 90° (π/2) | 1 | 0 | undefined |

3. Key Identities

Pythagorean identity: sin²θ + cos²θ = 1

Dividing by cos²θ: tan²θ + 1 = sec²θ

Dividing by sin²θ: 1 + cot²θ = cosec²θ

Double angle formulae:

• sin 2A = 2 sin A cos A
• cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
• tan 2A = 2tan A/(1 − tan²A)

Addition formulae:

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

4. Solving Trigonometric Equations

Method:

1. Rearrange to sin/cos/tan = value
2. Find the principal value (inverse trig)
3. Use CAST diagram or symmetry to find all solutions in the given interval

CAST diagram: Reading anticlockwise from positive x-axis:

• All positive (0°–90°)
• Sine positive (90°–180°)
• Tangent positive (180°–270°)
• Cosine positive (270°–360°)

Example: Solve 2sin²θ − sinθ − 1 = 0 for 0° ≤ θ ≤ 360°.

Let u = sinθ: 2u² − u − 1 = 0 → (2u + 1)(u − 1) = 0 → u = −1/2 or u = 1.

• sinθ = 1: θ = 90°
• sinθ = −1/2: θ = 210°, 330°

Solutions: θ = 90°, 210°, 330°

Example: Solve cos 2θ = sin θ for 0 ≤ θ ≤ π.

cos 2θ = 1 − 2sin²θ → 1 − 2sin²θ = sinθ → 2sin²θ + sinθ − 1 = 0.

(2sinθ − 1)(sinθ + 1) = 0 → sinθ = 1/2 or sinθ = −1.

In [0, π]: θ = π/6, 5π/6 (sinθ = 1/2); θ = 3π/2 outside range, sinθ = −1 → θ = 3π/2, not in [0,π].

Solutions: θ = π/6, 5π/6

5. The R Form: a sinθ + b cosθ = R sin(θ + α)

Express a sinθ + b cosθ in the form R sin(θ ± α) or R cos(θ ± α).

R = √(a² + b²), and α found from expansion and matching coefficients.

Example: Write 3sinθ + 4cosθ in the form R sin(θ + α).

R sin(θ + α) = R sinθ cosα + R cosθ sinα → R cosα = 3, R sinα = 4.

R = √(9+16) = 5; tanα = 4/3 → α = 53.1°.

3sinθ + 4cosθ = 5sin(θ + 53.1°)

Uses: finding maximum/minimum values, solving equations.

Maximum value = R (when sin = 1), minimum = −R.

6. Small Angle Approximations

For small θ (in radians):

• sin θ ≈ θ
• cos θ ≈ 1 − θ²/2
• tan θ ≈ θ

Example: Estimate sin(0.1). sin(0.1) ≈ 0.1 (actual: 0.0998 — very close).