Grade 12 Mathematics
Grade 12 · Term 2Mathematics

Trigonometry

We study compound angle identities, double angle formulae, and apply these to solving general trig equations and proving identities. We sketch trig graphs including $y=a\sin(bx+c)+d$.

Week 1

4.1 Compound & Double Angle Identities

  • Apply compound angle identities: $\sin(A\pm B)$, $\cos(A\pm B)$
  • Derive and apply double angle formulae: $\sin2A$, $\cos2A$
  • Solve trig equations for general solutions
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Real-World Connection

Compound angle formulas power signal processing. When two radio waves of different frequencies overlap, engineers use these exact formulas to analyse the resulting interference pattern — separating the components to recover the original signal.

Sine compound angle

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A\pm B) = \sin A\cos B \pm \cos A\sin B

Cosine compound angle

cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A\pm B) = \cos A\cos B \mp \sin A\sin B

Note: sign REVERSES for cosine

Double angle — sine

sin⁡2A=2sin⁡Acos⁡A\sin 2A = 2\sin A\cos A

Special case $B=A$ in the addition formula

Double angle — cosine (three forms)

cos⁡2A=cos⁡2A−sin⁡2A=2cos⁡2A−1=1−2sin⁡2A\cos 2A = \cos^2 A-\sin^2 A = 2\cos^2 A-1 = 1-2\sin^2 A

Use whichever form fits the context

💡 Tip

Memorise: sin compounds like 'sin-cos + cos-sin'. Cosine compounds like 'cos-cos − sin-sin' with the OPPOSITE sign to the bracket. Write it out phonetically: 'SC±CS' and 'CC∓SS'.

Worked Examples

Worked Example

Exact value using compound angles

Problem

Find the exact value of sin⁡75°\sin75° without a calculator.

Worked Example

General solution of trig equation

Problem

Solve sin⁡θ=32\sin\theta=\frac{\sqrt{3}}{2} for all solutions.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
Expand cos⁡(x−30°)\cos(x-30°) using compound angle formula.
2
L1 · Knowledge1 mark
Write sin⁡2A\sin2A in terms of sin⁡A\sin A and cos⁡A\cos A.
3
L2 · Routine Procedures3 marks
Find the exact value of cos⁡15°\cos15°.
4
L2 · Routine Procedures3 marks
Simplify sin⁡2x1+cos⁡2x\dfrac{\sin2x}{1+\cos2x}.
5
L3 · Complex Procedures5 marks
Prove cos⁡2θ−cos⁡4θ=2sin⁡3θsin⁡θ\cos2\theta-\cos4\theta=2\sin3\theta\sin\theta.
6
L3 · Complex Procedures5 marks
Solve cos⁡2x+sin⁡x=0\cos2x+\sin x=0 for x∈[0°,360°]x\in[0°,360°].
7
L4 · Problem Solving4 marks
If cos⁡α=−35\cos\alpha=-\frac{3}{5} (α\alpha in Q3), find sin⁡2α\sin2\alpha.
8
L4 · Problem Solving5 marks
Solve for general solution: 2cos⁡2θ−3cos⁡θ+1=02\cos^2\theta-3\cos\theta+1=0.

Next

Analytical Geometry

Trigonometry Grade 12 Maths CAPS Notes & Examples | MathSciBuddy