Grade 11 Maths · Trigonometry · Paper 2

Reduction Formulas Grade 11

Real Reduction Formulas past paper questions and answers from 2024 to 2025, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

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Question 3.1 · March 2025 · Provincial · 5 marks

3.1 Given that

13 cos β = 5

and

α + β = 9 0^{\circ}

. By making use of a sketch and WITHOUT the use of a calculator, calculate the value of

\frac{t a n β}{c o s α}

.(5)

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

Reduction Formulas skills

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More Reduction Formulas Grade 11 questions and answers

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4. Question 4.1.1 · November 2025 · Paper 2 · Provincial · 2 marks
4.1 In the diagram below, O is the point of origin. OR $= k$ , $R \hat{O} S = 45°$ , R $(t; 3)$ is a point in the first quadrant and point S lies on the $x$ -axis such that OS $⊥$ RS.

Without using a calculator, calculate the value(s) of the following: $Diagram for question 4.1$
4.1.1 $t$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Correct trig ratio $tan 45° = \frac{3}{t}$
  1 mark · Calculate a side using trig
2. ✓
  $t$ -value $1 = \frac{3}{t}$
  $t = 3$
  1 mark · Use special angle values
5. Question 2.1.1 · June 2024 · Paper 2 · Provincial · 2 marks
2.1 Given: $sin 2 0^{\circ} = p$ . Determine the following in terms of $p$ :
2.1.1 $sin (- 2 0^{\circ})$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Negative angle rule $sin (- 2 0^{\circ}) = - sin 2 0^{\circ}$
  1 mark · Use a reduction formula
2. ✓
  Answer in terms of p $= - p$
  1 mark · Use a reduction formula
6. Question 3.2.2 · June 2024 · Paper 2 · Provincial · 2 marks
3.2 If $3 tan β + 2 = 0$ and $sin β > 0$ , determine without the use of a calculator and with the aid of a sketch, the value of the following:
3.2.2 $13 cos β - 13 sin^{2} β$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Substitution $13 (- \frac{3}{13}) - 13 (\frac{2}{13})^{2}$
  1 mark · Find trig ratios from a sketch
2. ✓
  Answer $= - 3 - 4 = - 7$
  1 mark · Find trig ratios from a sketch
7. Question 5.1 · November 2024 · Paper 2 · Provincial · 5 marks
5.1 If $- 3 tan β - 5 = 0$ and $sin β < 0$ . Determine the value of $sin^{2} β - cos^{2} β$ with the aid of a diagram.(5)
Show the memo (5 ticks)Memo
1. ✓
  $tan β = \frac{- 5}{3}$ $- 3 tan β = 5$
  $tan β = \frac{- 5}{3}$
  1 mark · Solve a linear equation
2. ✓
  Diagram in the correct quadrant $tan < 0$ and $sin < 0$ : $β$ is in the 4th quadrant, with $x = 3$ , $y = - 5$ . $Memo diagram$
  1 mark · Decide the quadrant and sign
3. ✓
  r = $14$ $r = (3)^{2} + (- 5)^{2} = 14$
  1 mark · Use the theorem of Pythagoras
4. ✓
  Substitution $sin^{2} β - cos^{2} β = (\frac{- 5}{14})^{2} - (\frac{3}{14})^{2}$
  1 mark · Find trig ratios from a sketch
5. ✓
  Answer $= \frac{5}{14} - \frac{9}{14} = - \frac{2}{7}$
  1 mark · Find trig ratios from a sketch
8. Question 6.3 · November 2024 · Paper 2 · Provincial · 2 marks
The graph of $f (x) = cos (x + 30°)$ in the interval of $x \in [- 120°; 150°]$ has been drawn in the diagram below. $Diagram for this question$
6.3 Write down the equation of $h$ in its simplest form.(2)
Show the memo (2 ticks)Memo
1. ✓
  $cos (90° + x)$ $h (x) = cos (x + 60° + 30°) + 1 = cos (90° + x) + 1$
  1 mark · Shift a graph
2. ✓
  $- sin x$ $h (x) = - sin x + 1$
  1 mark · Use co-ratio identities
9. Question 5.2 · November 2024 · Paper 2 · Provincial · 6 marks
5.2 Simplify:

$\frac{t a n 135°. s i n 210°}{c o s 660° + c o s 180°}$ (6)
Show the memo (6 ticks)Memo
1. ✓
  $- tan 45°$ $tan 135° = - tan 45°$
  1 mark · Use a reduction formula
2. ✓
  $- sin 30°$ $sin 210° = - sin 30°$
  1 mark · Use a reduction formula
3. ✓
  $cos 60°$ $cos 660° = cos 300° = cos 60°$
  1 mark · Use a reduction formula
4. ✓
  $- 1$ $cos 180° = - 1$
  $= \frac{( - t a n 45° ) . ( - s i n 30° )}{c o s 60° - 1}$
  1 mark · Use a reduction formula
5. ✓
  Special angle values $= \frac{( - 1 ) ( - \frac{1}{2} )}{( \frac{1}{2} ) - 1}$
  1 mark · Use special angle values
6. ✓
  Answer $= - 1$
  1 mark · Use special angle values
10. Question 6.1 · November 2024 · Paper 2 · Provincial · 5 marks
6.1 Solve the following equation for $x \in [- 180°; 180°]$ .

$cos (x - 30°) = sin x$ (5)
Show the memo (5 ticks)Memo
1. ✓
  $sin x = cos (90° - x)$ $cos (x - 30°) = sin x$
  $cos (x - 30°) = cos (90° - x)$
  1 mark · Use co-ratio identities
2. ✓
  Equation with $k .360°$ $x - 30° = 90° - x + k .360°$ ; $k \in Z$
  1 mark · Write the general solution
3. ✓
  $x = 60° + k .180°$ $2 x = 120° + k .360°$
  $x = 60° + k .180°$
  1 mark · Write the general solution
4. ✓
  $k \in Z$ $k \in Z$
  1 mark · Write the general solution
5. ✓
  Solutions in the interval $x \in {- 120°; 60°}$
  1 mark · Find solutions in an interval

Reduction Formulas Grade 11 videos

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Special Triangles Grade 10: IntroductionKevinmathscience

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

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