MATHEMATICS INVESTIGATION: 2023
GRADE 12
NATIONAL SENIOR CERTIFICATE

INSTRUCTIONS AND INFORMATION
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.

INVESTIGATING COMPOUND ANGLES
QUESTION 1
1.1. In the following diagrams, ??? = ?, ??? = ?, ??? = ?, ?? ? = ?

Write each of the following in terms of α and β
1.1.1 ?? # ? _________________________ (1)
1.1.2 ?? # ? _________________________ (1)
1.2 Use your calculator to complete the table below. There is no need to show your working out.

(10)
1.2.2 What do you notice concerning the values of ???(? − ?) ??? ???? − ???? ?
(Hint – are the values the same or different?) (1)
1.2.3 What do you notice concerning the values of cos(? − β) and ???????? + ???????? (1)
1.2.4 What do you notice concerning the values of cos(? − β) and ???????? − ????????? (1)
1.2.5 Hence deduce a formula to expand cos(? − β) (2)
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SECTION B
QUESTION 2
2.1 Now let us investigate whether the identity of cos(? − β) = ???????? + ???????? is true for all values of α and β.
Let P (cos?; ????) and Q (cos?; ????) be any two points on the circle O with radius 1. If ??? = ? and ??? = ? then ??? = ? − ?

2.1.1 Make use of the cosine rule to determine the length of PQ. (4)
2.1.2 Make use of the distance formula to determine the length of PQ  (5)
2.1.3 Hence, compare number 2.1.1 and 2.1.2 and write a conclusion about cos(a – b). (3)
2.1.4 Use 2.1.3 [cos(? − ?) = ????. ???? + ????. ???? ] to derive a formula for cos(? + ?)
(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) (3)
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QUESTION 3
Applications
3.1 Express the following as single trigonometry ratio:
3.1.1 ???2?. ???3? − ???2?. ???3? (2)
3.1.2 ???2?. ???? + ???2?. ????  (2)
3.2 Determine the values of the following without using a calculator.
3.2.1 ???85°. ???25° − ???85°. ???25° (3)
3.2.2 ???160°. ???10° + ???160°. ???10° (4)
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