Wednesday, 08 February 2023 06:18

Mathematics Grade 12 Investigation 2023 Term 1

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Read the following instructions carefully before answering the questions.

  1. This task paper consists of 2 questions.
  2. Answer ALL the 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.

1.1. In the following diagrams, 𝐴𝐡𝐷 = 𝛽, 𝐷𝐡𝐢 = 𝛼, 𝐸𝐹𝐻 = 𝛽, 𝐸𝐹 𝐺 = 𝛼
1 1
Write each of the following in terms of Ξ± and Ξ²
1.1.1 𝐴𝐡 # 𝐢 _________________________ (1)
1.1.2 𝐻𝐹 # 𝐺 _________________________ (1)
1.2 Use your calculator to complete the table below. There is no need to show your working out.

Angles Β cos(𝛼 βˆ’ Ξ²) cos𝛼 βˆ’cosΞ² π‘π‘œπ‘ π›Όπ‘π‘œπ‘ π›½ + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 π‘π‘œπ‘ π›Ό cosΞ²βˆ’π‘ π‘–π‘›π›Όπ‘ π‘–π‘›π›½
𝛼 = 60ΒΊ
𝛽 = 30ΒΊ
cos(60Β° βˆ’ 30Β°)
= cos 30 = √3/2
cos 60 – cos 30 Β½Β Γ—Β βˆš3/2Β + √3/2Β Γ—Β Β½ Β½Β Γ—Β βˆš3/2Β -√3/2Β Γ—Β Β½
𝛼 = 110Β°
and Ξ²= 50Β°
Β  Β  Β  Β 
𝛼 = 87Β°
and Ξ²= 42Β°
Β  Β  Β  Β 
𝛼 = 223Β°
and Ξ² = 193Β°
Β  Β  Β  Β 

1.2.2 What do you notice concerning the values of π‘π‘œπ‘ (𝛼 βˆ’ 𝛽) π‘Žπ‘›π‘‘ π‘π‘œπ‘ π›Ό βˆ’ π‘π‘œπ‘ π›½ ?
(Hint – are the values the same or different?) (1)
1.2.3 What do you notice concerning the values of cos(𝛼 βˆ’ Ξ²) and π‘π‘œπ‘ π›Όπ‘π‘œπ‘ π›½ + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (1)
1.2.4 What do you notice concerning the values of cos(𝛼 βˆ’ Ξ²) and π‘π‘œπ‘ π›Όπ‘π‘œπ‘ π›½ βˆ’ 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽? (1)
1.2.5 Hence deduce a formula to expand cos(𝛼 βˆ’ Ξ²) (2)


2.1 Now let us investigate whether the identity of cos(𝛼 βˆ’ Ξ²) = π‘π‘œπ‘ π›Όπ‘π‘œπ‘ π›½ + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 is true for all values of Ξ± and Ξ².
Let P (cos𝛼; 𝑠𝑖𝑛𝛼) and Q (cos𝛽; 𝑠𝑖𝑛𝛽) be any two points on the circle O with radius 1. If 𝑃𝑂𝐴 = 𝛼 and 𝑄𝑂𝐴 = 𝛽 then 𝑃𝑂𝑄 = 𝛼 βˆ’ 𝛽
2 1
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 – b).Β (3)
2.1.4 Use 2.1.3 [cos(𝛼 βˆ’ 𝛽) = π‘π‘œπ‘ π›Ό. π‘π‘œπ‘ π›½ + 𝑠𝑖𝑛𝛼. 𝑠𝑖𝑛𝛽 ] to derive a formula for cos(𝛼 + 𝛽)
(Hint: use suitable reduction formula) (4)
2.1.5 Use cos(a – b) to derive a formula for sin(a – b).
(Hint: use co-function) (3)
2.1.6 Use cos(a – b) to derive a formula for sin(a + b).
(Hint: use co-function)Β (3)

3.1 Express the following as single trigonometry ratio:
3.1.1 π‘π‘œπ‘ 2π‘₯. π‘π‘œπ‘ 3π‘₯ βˆ’ 𝑠𝑖𝑛2π‘₯. 𝑠𝑖𝑛3π‘₯ (2)
3.1.2 𝑠𝑖𝑛2π‘₯. π‘π‘œπ‘ π‘₯ + π‘π‘œπ‘ 2π‘₯. 𝑠𝑖𝑛π‘₯Β Β (2)
3.2 Determine the values of the following without using a calculator.
3.2.1 𝑠𝑖𝑛85Β°. π‘π‘œπ‘ 25Β° βˆ’ π‘π‘œπ‘ 85Β°. 𝑠𝑖𝑛25Β° (3)
3.2.2 π‘π‘œπ‘ 160Β°. π‘π‘œπ‘ 10Β° + 𝑠𝑖𝑛160Β°. 𝑠𝑖𝑛10Β° (4)

Last modified on Wednesday, 08 February 2023 06:38