Grade 11 Maths · Trigonometry

Use the cosine rule

Use the cosine rule a² = b² + c² − 2bc cos A to find a side or angle, given three sides, or two sides and the angle between them.

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Question 7.1 · November 2024 · Paper 2 · Provincial · 2 marks

In the diagram below, circle ACD is given where AC is a diameter. DC is produced to B and AB = 8 units.

\hat{B} = 30°

and BC = 5 units. $Diagram for this question$

7.1 Prove that AC = 4,44 units (rounded off to TWO decimal places.)(2)

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More past paper questions: use the cosine rule

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3. 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
4. Question 10.2.4 · May/June 2025 · Paper 2 · National (DBE) · 3 marks
10.2 In the diagram, O is the centre of the circle. Points D and B lie on the circle. Points A and C lie outside the circle such that side AC of $Δ A D C$ passes through B. F is a point on BC such that $FO ∥ B D$ . $D R = RB$ and RO is drawn.
10.2.4 Calculate the size of $A \hat{B} D$ .(3)
Show the memo (1 tick)Memo
1. ✓
  METHOD 1: Using trigonometry in the right-angled triangle and exterior anglesIn right-angled $Δ D RO$ :
  $tan R \hat{D} O = \frac{RO}{R D} = \frac{4 , 5}{6} = \frac{3}{4}$
  $∴ R \hat{D} O = 36, 8 7^{\circ}$
  
  Given $F \hat{C} O = R \hat{D} O$ , so $F \hat{C} O = 36, 8 7^{\circ}$ (this is angle $\hat{C}$ )
  In $Δ B D C$ , $B D = BC$ , so $B \hat{D} C = \hat{C} = 36, 8 7^{\circ}$
  
  $A \hat{B} D$ is an exterior angle of $Δ B D C$ :
  $A \hat{B} D = B \hat{D} C + \hat{C}$ [ext $∠$ of $Δ$ ]
  $A \hat{B} D = 36, 8 7^{\circ} + 36, 8 7^{\circ}$
  $A \hat{B} D = 73, 7 4^{\circ}$
  3 marks · Use exterior angle of a triangle

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