Sine, Cosine & Area Rules (Grade 11) — Worksheet & Memo

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Grade 11 · Trigonometry

Sine, Cosine & Area Rules (Grade 11)

EXAM-STYLE CLASS TEST

Marks

Duration

1 hour 15 minutes

Questions

Name:

Class:

Date:

Mark

/ 50

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
The SINE rule is the one to use when you know …
(1)
A)two angles and a side
B)two sides and the angle BETWEEN them
C)all three sides
D)all three angles
1.2
The COSINE rule a² = b² + c² − 2bc cos A is used when …
(1)
A)two sides and the INCLUDED angle are known
B)two angles and a side are known
C)only one side is known
D)the triangle is right-angled
1.3
In △ABC, A = 40°, B = 60° and a = 10. Then b = …
(1)
A)13,47
B)7,42
C)8,66
D)15,32
1.4
In △PQR, p = 5, q = 7 and R = 60°. Then r = …
(1)
A)6,24
B)39
C)10,44
D)2
1.5
The AREA rule states that the area of a triangle is …
(1)
A)12ab sin C
B)12ab cos C
C)ab sin C
D)12 × base × height only
1.6
Find the area of △ABC if b = 8, c = 6 and A = 30°.
(1)
A)12
B)24
C)20,78
D)48
1.7
In △ABC, a = 9, b = 7 and c = 5. Then cos A = …
(1)
A)−0,1
B)0,1
C)−7
D)0,83
1.8
Hence, in that triangle, ∠A = …
(1)
A)95,74°
B)84,26°
C)5,74°
D)1,67°
1.9
The AMBIGUOUS case of the sine rule can arise when …
(1)
A)two sides and a NON-included angle are given
B)all three sides are given
C)two angles and a side are given
D)one of the angles is a right angle
1.10
In △ABC, if C = 90° the cosine rule becomes …
(1)
A)c² = a² + b²
B)c² = a² − b²
C)c² = a² + b² − 2ab
D)c = a + b

Answer grid — circle your answers for Question 1

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[12 MARKS]

In the diagram below, △ABC is drawn with BÂC = α, AB̂C = β and AB = c.

2.1
Write down the size of AĈB in terms of α and β.
(1)
2.2
Show that BC = c · sin αsin(α + β).
(4)
2.3
Given that α = 52°, β = 61° and c = 48 m, calculate the length of BC, correct to TWO decimal places.
(3)
2.4
Hence calculate the area of △ABC, correct to TWO decimal places.
(4)

Question 3

[15 MARKS]

In the diagram below, ABCD is a quadrilateral with the diagonal AC drawn. AB̂C = 106°, BÂC = 31°, AĈD = 46°, AC = 4,6 cm and CD = 10 cm. BC = x. The diagram is not drawn to scale.

3.1
Calculate the length of x, correct to TWO decimal places.
(3)
3.2
Calculate the area of △ABC, correct to TWO decimal places.
(4)
3.3
Calculate the length of AD, correct to TWO decimal places.
(4)
3.4
Hence calculate the area of quadrilateral ABCD, correct to TWO decimal places.
(4)

Question 4

[13 MARKS]

VABC is a pyramid with V the apex and △ABC its horizontal base. BÂC = 110°, AB̂C = 40° and BC = 6 m. The perpendicular height of the pyramid is 8 m. [Volume of a pyramid = ⅓ × area of the base × perpendicular height]

4.1
Write down the size of AĈB, and hence calculate the length of AB, correct to TWO decimal places.
(4)
4.2
Hence calculate the area of the base △ABC, correct to TWO decimal places.
(3)
4.3
Hence calculate the volume of the pyramid, correct to TWO decimal places.
(3)
4.4
Explain why the AREA RULE had to be used for the base rather than ½ × base × perpendicular height.
(3)

TOTAL: 50 marks

This question paper consists of 4 questions.

Statement	Reason

Statement	Reason