Special Angles, Reciprocal Ratios & Trig Equations (Grade 10)

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Grade 10 · Trigonometry

MARKING GUIDELINE

Marks

Duration

1 hour

Questions

Name:

Class:

Date:

Mark

/ 37

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
sin 30° = …
(1)
A)12✓
B)√32
C)1√2
D)1
Answer: A — In the 30°-60°-90° triangle the side opposite 30° is half the hypotenuse.
- B — that is cos 30°
- C — that is sin 45°
- D — that is sin 90°
1.2
tan 60° = …
(1)
A)√3✓
B)1√3
C)√32
D)1
Answer: A — tan 60° = opposite ÷ adjacent = √3 ÷ 1.
- B — that is tan 30° — the ratio is inverted
- C — that is sin 60°
- D — that is tan 45°
1.3
cos 0° = …
(1)
A)1✓
B)0
C)12
D)undefined
Answer: A — At 0° the adjacent side and the hypotenuse coincide, so their ratio is 1.
- B — that is sin 0°
- C — that is cos 60°
- D — cos 0° is perfectly well defined; it is tan 90° that is not
1.4
tan 90° is …
(1)
A)undefined✓
B)0
C)1
D)∞
Answer: A — tan 90° would need a division by an adjacent side of zero, so it has no value.
- B — that is tan 0°
- C — that is tan 45°
- D — ∞ is not a value a ratio can take — the correct word is undefined
1.5
sec θ = …
(1)
A)1cos θ✓
B)1sin θ
C)1tan θ
D)cos θ
Answer: A — Secant is the reciprocal of COSINE.
- B — that is cosec θ
- C — that is cot θ
- D — that is the ratio itself, not its reciprocal
1.6
cot 45° = …
(1)
A)1✓
B)0
C)√3
D)1√2
Answer: A — cot 45° = 1 ÷ tan 45° = 1 ÷ 1 = 1.
- B — that is cot 90°
- C — that is cot 30°
- D — that is cos 45°, not its cotangent
1.7
Solve for θ if 2 sin θ = 1 and 0° ≤ θ ≤ 90°.
(1)
A)30°✓
B)60°
C)45°
D)0,5°
Answer: A — sin θ = 0,5, and sin 30° = 0,5.
- B — that is the angle whose COSINE is 0,5
- C — sin 45° = 1/√2, not 0,5
- D — wrote the ratio down as though it were the angle
1.8
Solve for θ if tan θ = √3 and 0° ≤ θ ≤ 90°.
(1)
A)60°✓
B)30°
C)45°
D)√3°
Answer: A — tan 60° = √3.
- B — tan 30° = 1/√3, the reciprocal
- C — tan 45° = 1
- D — wrote the ratio down as though it were the angle
1.9
sin 60° × cos 30° = …
(1)
A)34✓
B)14
C)√34
D)√32
Answer: A — Both equal √3/2, and (√3/2)(√3/2) = 3/4.
- B — used sin 30° × cos 60° instead
- C — multiplied out only one of the two √3 factors
- D — copied one of the two values instead of multiplying them
1.10
Solve for θ if 3 tan θ − 3 = 0 and 0° ≤ θ ≤ 90°.
(1)
A)45°✓
B)60°
C)30°
D)1°
Answer: A — 3 tan θ = 3 gives tan θ = 1, and tan 45° = 1.
- B — tan 60° = √3, not 1
- C — tan 30° = 1/√3, not 1
- D — wrote the ratio tan θ = 1 down as though it were the angle

Answer grid — marking guideline

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

[15 MARKS]

Answer the questions below WITHOUT the use of a calculator. Leave your answers in simplest surd form where necessary.

2.1
Evaluate: sin 30°·cos 60° + tan 45°
(4)
= 12 × 12 + 1 (2)
= 14 + 1 (1) = 54 (1)
2.2
Evaluate: tan² 60° − sin 30°
(4)
tan 60° = 3, so tan² 60° = 3 (2)
3 − 12 (1) = 52 (1)
2.3
Evaluate: cosec 30° + sec 60°
(4)
cosec 30° = 1sin 30° = 2 (2)
sec 60° = 1cos 60° = 2 (1)
2 + 2 = 4 (1)
2.4
Show that cos² 45° + sin² 45° = 1.
(3)
cos 45° = sin 45° = 22 (1)
24 + 24 (1) = 1 (1)

Question 3

[12 MARKS]

Solve for θ if θ ∈ [0° ; 90°]. Show ALL your working.

3.1
2 cos θ − 1 = 0
(3)
cos θ = 12 (2)
θ = 60° (1)
3.2
3 tan θ = 3
(4)
tan θ = 33 = 13 (2)
θ = tan⁻¹(13) (1) = 30° (1)
3.3
4 sin θ − 3 = 0, correct to ONE decimal place
(3)
sin θ = 0,75 (1)
θ = sin⁻¹(0,75) (1) = 48,6° (1)
3.4
Explain why tan 90° is undefined, but sin 90° is not.
(2)
tan θ = sin θcos θ and cos 90° = 0 (1), so the division is impossible; sin 90° is simply 1 (1)

TOTAL: 37 marks

This question paper consists of 3 questions.