Wednesday, 08 February 2023 06:18

## Mathematics Grade 12 Investigation 2023 Term 1

MATHEMATICS INVESTIGATION: 2023
NATIONAL SENIOR CERTIFICATE

INSTRUCTIONS AND INFORMATION

1. This task paper consists of 2 questions.
3. Number the answers correctly according to the numbering system used in this question paper
4. Clearly show ALL calculations, diagrams, graphs, et cetera which you have used in determining your answers.
5. Answers only will not necessarily be awarded full marks.
6. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
7. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise.
8. Diagrams are NOT necessarily drawn to scale.
9. Write neatly and legibly.

INVESTIGATING COMPOUND ANGLES
QUESTION 1
1.1. In the following diagrams, π΄π΅π· = π½, π·π΅πΆ = πΌ, πΈπΉπ» = π½, πΈπΉ πΊ = πΌ

Write each of the following in terms of Ξ± and Ξ²
1.1.1 π΄π΅ # πΆ _________________________ (1)
1.1.2 π»πΉ # πΊ _________________________ (1)
1.2 Use your calculator to complete the table below. There is no need to show your working out.

 Angles Β cos(πΌ β Ξ²) cosπΌ βcosΞ² πππ πΌπππ π½ + π πππΌπ πππ½ πππ πΌ cosΞ²βπ πππΌπ πππ½ ππ:πΌ = 60ΒΊπ½ = 30ΒΊ cos(60Β° β 30Β°)= cos 30 = β3/2 cos 60 β cos 30 Β½Β ΓΒ β3/2Β +Β β3/2Β ΓΒ Β½ Β½Β ΓΒ β3/2Β -β3/2Β ΓΒ Β½ πΌ = 110Β°and Ξ²= 50Β° Β Β Β Β πΌ = 87Β°and Ξ²= 42Β° Β Β Β Β πΌ = 223Β°and Ξ² = 193Β° Β Β Β Β

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1.2.2 What do you notice concerning the values of πππ (πΌ β π½) πππ πππ πΌ β πππ π½ ?
(Hint β are the values the same or different?) (1)
1.2.3 What do you notice concerning the values of cos(πΌ β Ξ²) and πππ πΌπππ π½ + π πππΌπ πππ½ (1)
1.2.4 What do you notice concerning the values of cos(πΌ β Ξ²) and πππ πΌπππ π½ β π πππΌπ πππ½? (1)
1.2.5 Hence deduce a formula to expand cos(πΌ β Ξ²) (2)
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SECTION B
QUESTION 2

2.1 Now let us investigate whether the identity of cos(πΌ β Ξ²) = πππ πΌπππ π½ + π πππΌπ πππ½ is true for all values of Ξ± and Ξ².
Let P (cosπΌ; π πππΌ) and Q (cosπ½; π πππ½) be any two points on the circle O with radius 1. If πππ΄ = πΌ and πππ΄ = π½ then πππ = πΌ β π½

2.1.1 Make use of the cosine rule to determine the length of PQ.Β (4)
2.1.2 Make use of the distance formula to determine the length of PQΒ Β (5)
2.1.3 Hence, compare number 2.1.1 and 2.1.2 and write a conclusion about cos(a β b).Β (3)
2.1.4 Use 2.1.3 [cos(πΌ β π½) = πππ πΌ. πππ π½ + π πππΌ. π πππ½ ] to derive a formula for cos(πΌ + π½)
(Hint: use suitable reduction formula) (4)
2.1.5 Use cos(a β b) to derive a formula for sin(a β b).
(Hint: use co-function) (3)
2.1.6 Use cos(a β b) to derive a formula for sin(a + b).
(Hint: use co-function)Β (3)
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QUESTION 3
Applications
3.1 Express the following as single trigonometry ratio:
3.1.1 πππ 2π₯. πππ 3π₯ β π ππ2π₯. π ππ3π₯ (2)
3.1.2 π ππ2π₯. πππ π₯ + πππ 2π₯. π πππ₯Β Β (2)
3.2 Determine the values of the following without using a calculator.
3.2.1 π ππ85Β°. πππ 25Β° β πππ 85Β°. π ππ25Β° (3)
3.2.2 πππ 160Β°. πππ 10Β° + π ππ160Β°. π ππ10Β° (4)
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