Trigonometry
What the exam tests
The paper checks fluency with radian measure, allied angles, compound and multiple-angle identities, and simple trigonometric equations. Business contexts include periodic models, but most marks come from identity manipulation.
Core rules
- π radians = 180°; arc length s = rθ with θ in radians; sector area = ½r²θ.
- ASTC: in quadrants I, II, III, IV, the positive functions are All, Sin, Tan, Cos respectively.
- sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB.
- sin2A = 2sinA cosA; cos2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1.
- tan(A + B) = (tanA + tanB)/(1 − tanA tanB).
Worked example
Find sin 75° without a calculator. 75° = 45° + 30°, so sin75° = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4. Check numerically: (2.449 + 1.414)/4 ≈ 0.966, matching sin75° ≈ 0.9659.
Also, sin75° + sin15° = 2 sin45° cos30° = 2(√2/2)(√3/2) = √6/2, using sum-to-product.
Traps
- Using degrees in s = rθ; θ must be in radians.
- Wrong sign in cos(A+B) = cosA cosB − sinA sinB (the sign reverses).
- Writing sin2A = 2sinA instead of 2sinA cosA.
- Ignoring the quadrant when solving for an angle, and so losing a valid solution.
- Confusing sin²A with sin(A²).