## Mathematics grade 12 investigation 2023 memorandum

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Mathematics grade 12 investigation 2023 memorandum; Mathematics Department of the Education Republic of the Philippines This book was collaboratively developed and reviewed. Mathematics, arithmetic, measurement, geometry, fractions, and more. This course is aligned with Common Core standards. (is an area of knowledge that includes such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes)

Mathematics grade 12 investigation 2023 memorandum; is a student & teacher-friendly website compiling the entire grade 10 math curriculum? It includes interactive quizzes and video tutorials and we provide a comprehensive collection of National, Western Cape (WC), Kwa-Zulu Natal (KZN), Gauteng (GP), Eastern Cape (EC), Mpumalanga (MP), North West (NW), and Free State (FS).

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## Sample Paper

**MATHEMATICS INVESTIGATION: 2023**

GRADE 12

NATIONAL SENIOR CERTIFICATE

**INSTRUCTIONS AND INFORMATION**

Read the following instructions carefully before answering the questions.

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

**INVESTIGATING COMPOUND ANGLES**

**QUESTION 1**

1.1. In the following diagrams, π΄π΅π· = π½, π·π΅πΆ = πΌ, πΈπΉπ» = π½, πΈπΉ πΊ = πΌ

Write each of the following in terms of Ξ± and Ξ²

1.1.1 π΄π΅ # πΆ _________________________ (1)

1.1.2 π»πΉ # πΊ _________________________ (1)

1.2 Use your calculator to complete the table below. There is no need to show your working out.

Angles | Β cos(πΌ β Ξ²) | cosπΌ βcosΞ² | πππ πΌπππ π½ + π πππΌπ πππ½ | πππ πΌ cosΞ²βπ πππΌπ πππ½ |

ππ: πΌ = 60ΒΊ π½ = 30ΒΊ |
cos(60Β° β 30Β°) = cos 30 =Β ^{β3}/_{2} |
cos 60 β cos 30 | Β½Β ΓΒ ^{β3}/_{2Β }+Β ^{β3}/_{2}Β ΓΒ Β½ |
Β½Β ΓΒ ^{β3}/_{2Β }–^{β3}/_{2}Β ΓΒ Β½ |

πΌ = 110Β° and Ξ²= 50Β° |
||||

πΌ = 87Β° and Ξ²= 42Β° |
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πΌ = 223Β° and Ξ² = 193Β° |

(10)

1.2.2 What do you notice concerning the values of πππ (πΌ β π½) πππ πππ πΌ β πππ π½ ?

(Hint β are the values the same or different?) (1)

1.2.3 What do you notice concerning the values of cos(πΌ β Ξ²) and πππ πΌπππ π½ + π πππΌπ πππ½ (1)

1.2.4 What do you notice concerning the values of cos(πΌ β Ξ²) and πππ πΌπππ π½ β π πππΌπ πππ½? (1)

1.2.5 Hence deduce a formula to expand cos(πΌ β Ξ²) (2)

**[17]
**

**SECTION B
QUESTION 2**

2.1 Now let us investigate whether the identity of cos(πΌ β Ξ²) = πππ πΌπππ π½ + π πππΌπ πππ½ is true for all values of Ξ± and Ξ².

Let P (cosπΌ; π πππΌ) and Q (cosπ½; π πππ½) be any two points on the circle O with radius 1. If πππ΄ = πΌ and πππ΄ = π½ then πππ = πΌ β π½

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(πΌ β π½) = πππ πΌ. πππ π½ + π πππΌ. π πππ½ ] to derive a formula for cos(πΌ + π½)

(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)

**[22]**

**QUESTION 3**

Applications

3.1 Express the following as single trigonometry ratio:

3.1.1 πππ 2π₯. πππ 3π₯ β π ππ2π₯. π ππ3π₯ (2)

3.1.2 π ππ2π₯. πππ π₯ + πππ 2π₯. π πππ₯Β Β (2)

3.2 Determine the values of the following without using a calculator.

3.2.1 π ππ85Β°. πππ 25Β° β πππ 85Β°. π ππ25Β° (3)

3.2.2 πππ 160Β°. πππ 10Β° + π ππ160Β°. π ππ10Β° (4)

**[11]**

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