Topic 5 of 9 · Form 3
Circular measure and trigonometry
π rad = 180°; arc length s = rθ; sector area A = ½r²θ.
Notes for this topic 1 file, in order
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Radians
π rad = 180°; arc length s = rθ; sector area A = ½r²θ.
Worked example: A sector has radius 8 cm and angle 1.2 rad. Find its perimeter and area.
Arc = 9.6 cm; perimeter = 9.6 + 16 = 25.6 cm; area = ½ × 64 × 1.2 = 38.4 cm²
Trigonometric functions
Graphs of sin, cos and tan (amplitude, period); functions of the form a sin(bx) + c; the six functions including sec, cosec and cot.
Identities
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ; tan θ = sin θ/cos θ. Prove identities by working on one side.
Equations
Solve within a given range, in degrees or radians, including compound angles such as sin(2x + 30°) = 0.5.
Worked example: Solve 2cos²x + 3 sin x − 3 = 0 for 0° ≤ x ≤ 360°.
2(1 − sin²x) + 3 sin x − 3 = 0 → 2sin²x − 3 sin x + 1 = 0 → (2 sin x − 1)(sin x − 1) = 0
sin x = ½: x = 30°, 150°; sin x = 1: x = 90°
Check yourself
- Convert 3π/4 radians to degrees.
- State the amplitude and period of y = 3 sin 2x.
- Prove that (1 − cos²θ)/cos²θ = tan²θ.
Show answers
- 135°
- Amplitude 3; period 180° (π)
- sin²θ/cos²θ = tan²θ
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