Grade 12 Maths · Trigonometry · Paper 2

Compound Angles Grade 12

Real Compound Angles past paper questions and answers from 2022 to 2025, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

past paper questions: 21 past paper questions
skills: 3 skills
videos: 3 videos

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Question 6.1 · May/June 2025 · Paper 2 · National (DBE) · 4 marks

6.1 Prove that

2 cos^{2} (4 5^{\circ} + x) = 1 - sin 2 x

.(4)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

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More Compound Angles Grade 12 questions and answers

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4. Question 5.1.3 · May/June 2025 · Paper 2 · National (DBE) · 4 marks
5.1 If $cos θ = - \frac{5}{13}$ where $18 0^{\circ} < θ < 36 0^{\circ}$ , determine, without using a calculator, the value of:
5.1.3 $cos (θ - 13 5^{\circ})$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Expand using the compound angle identity for cosine $cos (θ - 13 5^{\circ}) = cos θ cos 13 5^{\circ} + sin θ sin 13 5^{\circ}$
  1 mark · Use a compound angle identity
2. ✓
  Reduce angles to special angles $= cos θ (- cos 4 5^{\circ}) + sin θ (sin 4 5^{\circ})$
  1 mark · Use a reduction formula
3. ✓
  Substitute trigonometric values $= (- \frac{5}{13}) (- \frac{2}{2}) + (- \frac{12}{13}) (\frac{2}{2})$
  
  OR
  
  $= (- \frac{5}{13}) (- \frac{1}{2}) + (- \frac{12}{13}) (\frac{1}{2})$
  1 mark · Find trig ratios from a sketch
4. ✓
  Simplify to final answer $= \frac{5 2}{26} - \frac{12 2}{26}$
  $= - \frac{7 2}{26}$
  1 mark · Find trig ratios from a sketch
5. Question 5.1.3 · March 2025 · Provincial · 4 marks
5.1 If $5 sin β - 4 = 0$ and $β \in (9 0^{\circ}; 27 0^{\circ})$ , determine without the use of a calculator and with the aid of a diagram:
5.1.3 $sin 3 β$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Expand compound angle $s in 3 b e t a = s in (2 b e t a + b e t a) = s in 2 b e t a cos b e t a + cos 2 b e t a s in b e t a$
  1 mark · Use a compound angle identity
2. ✓
  Expand double angle terms $= (2 sin β cos β) cos β + cos 2 β sin β$
  1 mark · Use a double angle identity
3. ✓
  Substitute values $= 2 (\frac{4}{5}) (- \frac{3}{5}) (- \frac{3}{5}) + (- \frac{7}{25}) (\frac{4}{5})$
  $= \frac{72}{125} - \frac{28}{125}$
  1 mark · Find trig ratios from a sketch
4. ✓
  Calculate result $= \frac{44}{125}$
  1 mark · Find trig ratios from a sketch
6. Question 6.1.3 · March 2024 · Paper 1 · Provincial · 3 marks
6.1 Given: $tan x = \frac{3}{4}$ where $x \in [18 0^{\circ}; 27 0^{\circ}]$ . With the aid of a sketch, and without the use of a calculator, calculate:
6.1.3 $cos^{2} (9 0^{\circ} - x) - 1$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Use co-ratio reduction $sin^{2} x - 1$
  1 mark · Use co-ratio identities
2. ✓
  Substitute and solve $(\frac{- 3}{5})^{2} - 1 = \frac{9}{25} - 1$
  1 mark · Find trig ratios from a sketch
3. ✓
  Final result $- \frac{16}{25}$
  1 mark · Find trig ratios from a sketch
7. Question 7.1.1 · March 2024 · Paper 1 · Provincial · 3 marks
7.1 Given: $cos (A - B) = cos A cos B + sin A sin B$
7.1.1 Use the above identity to deduce that $sin (A + B) = sin A cos B + cos A sin B$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Use co-ratios to convert sine to cosine $sin (A + B) = cos [9 0^{\circ} - (A + B)]$
  $= cos [(9 0^{\circ} - A) - B]$
  1 mark · Use co-ratio identities
2. ✓
  Apply given cosine difference identity $cos (9 0^{\circ} - A) cos B + sin (9 0^{\circ} - A) sin B$
  1 mark · Use a compound angle identity
3. ✓
  Convert back using co-ratios $sin A cos B + cos A sin B$
  1 mark · Use co-ratio identities
8. Question 6.1.2 · March 2022 · Provincial · 6 marks
6.1 Given $cos 2 0^{\circ} = p$ and $sin 1 4^{\circ} = q$ .

Without using a calculator, calculate the value of the following in terms of $p$ , $q$ or $p$ and $q$ .
6.1.2 $cos 6^{\circ}$ (6)
Show the memo (4 ticks)Memo
1. ✓
  Express as difference $cos 6^{\circ} = cos (2 0^{\circ} - 1 4^{\circ})$
  1 mark · Use a compound angle identity
2. ✓
  Expand $= cos 2 0^{\circ} cos 1 4^{\circ} + sin 2 0^{\circ} sin 1 4^{\circ}$
  1 mark · Use a compound angle identity
3. ✓
  Determine cos 14 $cos 1 4^{\circ} = 1 - q^{2}$
  1 mark · Use the square identity
4. ✓
  Substitute and solve $= p (1 - q^{2}) + (1 - p^{2}) (q)$
  3 marks · Find trig ratios from a sketch
9. Question 7.2 · March 2022 · Provincial · 5 marks
7.2 Calculate, without the use of a calculator, the value of: $\frac{2 s i n ^{2} 22 , 5 ^{\circ} - 1}{4 s i n 22 , 5 ^{\circ} c o s 22 , 5 ^{\circ}}$ (5)
Show the memo (4 ticks)Memo
1. ✓
  Rewrite numerator $2 sin^{2} 22, 5^{\circ} - 1 = - (1 - 2 sin^{2} 22, 5^{\circ}) = - cos (2 \times 22, 5^{\circ}) = - cos 4 5^{\circ}$
  2 marks · Use a double angle identity
2. ✓
  Rewrite denominator $4 sin 22, 5^{\circ} cos 22, 5^{\circ} = 2 (2 sin 22, 5^{\circ} cos 22, 5^{\circ}) = 2 sin 4 5^{\circ}$
  1 mark · Use a double angle identity
3. ✓
  Substitute values $\frac{- \frac{1}{2}}{2 ( \frac{1}{2} )}$
  1 mark · Use special angle values
4. ✓
  Calculate final answer $= - \frac{1}{2}$
  1 mark · Use special angle values
10. Question 4.1.2 · March 2022 · Provincial · 3 marks
4.1 Given $sin 2 4^{\circ} = m$ and $cos 3 5^{\circ} = n$ . Determine the following in terms of $m$ or $n$ .
4.1.2 $sin 7 0^{\circ}$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Use double angle identity $sin 7 0^{\circ} = sin (2 \times 3 5^{\circ}) = 2 sin 3 5^{\circ} cos 3 5^{\circ}$
  1 mark · Use a double angle identity
2. ✓
  Determine sin 35Since $cos 3 5^{\circ} = n$ , $sin 3 5^{\circ} = 1 - n^{2}$
  1 mark · Use the square identity
3. ✓
  Substitute and solve $2 (1 - n^{2}) (n) = 2 n 1 - n^{2}$
  1 mark · Use a double angle identity

Compound Angles Grade 12 videos

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Grade 11 Trigonometry Reduction with Co-FunctionsMiss Martins Maths and Science

Compound Angles Grade 12Kevinmathscience

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