Grade 12 Maths · Trigonometry · Paper 2

2D and 3D Trigonometry Grade 12

Real 2D and 3D Trigonometry 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: 20 past paper questions
skills: 5 skills
videos: 5 videos

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Question 8.1 · May/June 2025 · Paper 2 · National (DBE) · 2 marks

In the diagram, A, B and C lie in the same horizontal plane with

A B = A C

. D is directly above A such that

2 A D = BC

. Also,

A D = p

A \hat{B} C = x

and

D \hat{B} A = y

8.1 Determine AB in terms of p and y.(2)

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

2D and 3D Trigonometry skills

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More 2D and 3D Trigonometry Grade 12 questions and answers

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4. Question 5.1.1 · May/June 2025 · Paper 2 · National (DBE) · 3 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:
5.1.1 $sin^{2} θ$ (3)
Show the memo (2 ticks)Memo
1. ✓
  METHOD 1: Calculate the y-value using Pythagoras $cos θ = \frac{x}{r} = \frac{- 5}{13}$
  $x = - 5$ , $r = 13$
  $y^{2} = r^{2} - x^{2}$
  $y^{2} = 1 3^{2} - (- 5)^{2}$
  $y^{2} = 169 - 25 = 144$
  $y = \pm 12$
  Since $18 0^{\circ} < θ < 36 0^{\circ}$ and $cos θ$ is negative, $θ$ is in the 3rd quadrant. Therefore, $y = - 12$ .
  1 mark · Use the theorem of Pythagoras
2. ✓
  Calculate the squared sine ratio $sin^{2} θ = (\frac{y}{r})^{2}$
  $sin^{2} θ = (- \frac{12}{13})^{2}$
  $sin^{2} θ = \frac{144}{169}$
  2 marks · Find trig ratios from a sketch
5. Question 5.3.1 · March 2025 · Provincial · 2 marks
5.3 A piece of land has the form of a quadrilateral ABCD with $A B = 20 m$ , $BC = 12 m$ and $C D = 7 m$ . $\hat{B} = 11 0^{\circ}$ and $\hat{C D A} = 7 1^{\circ}$ . The owner decides to divide the land into two plots by erecting a fence from A to C.
5.3.1 Calculate the length of the fence AC correct to one decimal place.(2)
Show the memo (2 ticks)Memo
1. ✓
  Apply Cosine Rule $A C^{2} = 1 2^{2} + 2 0^{2} - 2 (12) (20) cos 11 0^{\circ}$
  1 mark · Use the cosine rule
2. ✓
  Calculate length $A C^{2} = 708, 17...$
  $A C = 26, 61 m$
  1 mark · Use the cosine rule
6. Question 5.1.1 · March 2025 · Provincial · 3 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:
5.1.1 $cos β$ (3)
Show the memo (3 ticks)Memo
1. ✓
  $sin β = \frac{4}{5}$ $sin β = \frac{4}{5}$ , so $y = 4$ , $r = 5$ .
  $sin β > 0$ and $β \in (9 0^{\circ}; 27 0^{\circ})$ , so $β$ is in the second quadrant.
  1 mark · Find trig ratios from a sketch
2. ✓
  x-value $x^{2} = r^{2} - y^{2} = 5^{2} - 4^{2} = 9$
  $x = - 3$ (second quadrant)
  1 mark · Use the theorem of Pythagoras
3. ✓
  Write ratio $cos β = \frac{x}{r} = - \frac{3}{5}$
  1 mark · Find trig ratios from a sketch
7. Question 6.1.1 · 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:
6.1.1 $sin x$ (3)
Show the memo (2 ticks)Memo
1. ✓
  Draw sketch in 3rd quadrant and calculate hypotenuse $x = - 4, y = - 3$
  $r^{2} = (- 4)^{2} + (- 3)^{2} = 25$
  $r = 5$
  2 marks · Use the theorem of Pythagoras
2. ✓
  State ratio $sin x = \frac{- 3}{5}$
  1 mark · Find trig ratios from a sketch
8. Question 5.1 · March 2022 · Provincial · 4 marks
A mountain climber wants to determine the height $PQ$ of a mountain. The climber is standing at $R$ on a flat ground. $R$ and $S$ are in the same horizontal plane as the foot of the mountain $Q$ . From $R$ , he measures the following angles: The angle of elevation of the top of the mountain $P$ is $w^{\circ}$ . $P \hat{R} S$ is $x^{\circ}$ . He then walks $z$ metres to point $S$ and measures $R \hat{S} P$ which is $y^{\circ}$ .
5.1 Show that $PQ = \frac{z s i n y \cdot s i n w}{s i n ( x + y )}$ (4)
Show the memo (3 ticks)Memo
1. ✓
  Calculate angle RPS in triangle PRS $R \hat{P} S = 18 0^{\circ} - (x + y)$
  1 mark · Use angle sum of a triangle
2. ✓
  Apply Sine Rule in triangle PRS $\frac{RP}{s i n y} = \frac{z}{s i n ( 18 0 ^{\circ} - ( x + y ))}$
  $RP = \frac{z s i n y}{s i n ( x + y )}$
  2 marks · Use the sine rule
3. ✓
  Calculate PQ in triangle PQR $\frac{PQ}{PR} = sin w$
  $PQ = PR sin w$
  $PQ = \frac{z s i n y \cdot s i n w}{s i n ( x + y )}$
  1 mark · Calculate a side using trig
9. Question 2.3.1 · March 2022 · Provincial · 3 marks
2.3 If $5 cos A + 3 = 0$ and $A \in [0^{\circ}; 18 0^{\circ}]$ , determine WITHOUT the use of calculator:
2.3.1 $sin A$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Determine cos A and quadrant $cos A = - \frac{3}{5}$
  Since $A \in [0^{\circ}; 18 0^{\circ}]$ and $cos A$ is negative, A is in Quadrant 2.
  1 mark · Decide the quadrant and sign
2. ✓
  Calculate y $x = - 3, r = 5$
  $x^{2} + y^{2} = r^{2} ⟹ (- 3)^{2} + y^{2} = 5^{2}$
  $y^{2} = 16 ⟹ y = 4$ (positive in Q2)
  1 mark · Use the theorem of Pythagoras
3. ✓
  Determine sin A $sin A = \frac{y}{r} = \frac{4}{5}$
  1 mark · Find trig ratios from a sketch
10. Question 4.1 · March 2022 · Provincial · 2 marks
In the diagram below, A, B and C are in the same horizontal plane. P is a point vertically above A. The angle of elevation from B to P is $α$ . $A \hat{C} B = β$ , $A \hat{B} C = θ$ and $BC = 20$ units.
4.1 Write AP in terms of AB and $α$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Use tan ratio in right-angled triangle PAB $tan α = \frac{A P}{A B}$
  1 mark · Calculate a side using trig
2. ✓
  Make AP subject $A P = A B tan α$
  1 mark · Calculate a side using trig

2D and 3D Trigonometry Grade 12 videos

Short lessons from South African teachers first. Videos load only when you tap play, to save your data.

Trig Triangles Grade 10: Find angleKevinmathscience

Sine Rule grade 11: IntroductionKevinmathscience

Cos Rule grade 11: IntroductionKevinmathscience

Area Rule grade 11: IntroductionKevinmathscience

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