Trigonometry: Solving Triangles & Angles of Elevation — Worksheet & Memo

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Grade 10 · Trigonometry

Trigonometry: Solving Triangles & Angles of Elevation

MARKING GUIDELINE

Marks

Duration

45 minutes

Questions

Name:

Class:

Date:

Mark

/ 30

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
An angle of elevation is measured …
(1)
A)from the horizontal UP to the line of sight✓
B)from the vertical down to the line of sight
C)from the line of sight up to the vertical
D)between the two objects
Answer: A — Angles of elevation and of depression are always measured from the HORIZONTAL.
- B — that angle is the COMPLEMENT of the angle of elevation
- C — the angle is measured FROM the horizontal, not TO the vertical
- D — an angle of elevation needs an observer and a horizontal, not two objects
1.2
A tower is 45 m tall. From a point 60 m away on level ground, the angle of elevation of its top is …
(1)
A)36,87°✓
B)53,13°
C)48,59°
D)0,64°
Answer: A — tan θ = 45/60 = 0,75, so θ = tan⁻¹(0,75) = 36,87°.
- B — inverted the ratio — that is the angle at the TOP of the tower
- C — used sin⁻¹(45 ÷ 60), but 60 m is the horizontal distance, not the line of sight
- D — the calculator was in RADIAN mode
1.3
From the top of a cliff 80 m high the angle of depression of a boat is 25°. The boat is … from the foot of the cliff.
(1)
A)171,56 m✓
B)37,30 m
C)189,30 m
D)72,50 m
Answer: A — The angle of depression equals the angle of elevation from the boat, so tan 25° = 80 ÷ d and d = 80 ÷ tan 25°.
- B — MULTIPLIED by tan 25° instead of dividing
- C — used 80 ÷ sin 25°, which gives the line of sight, not the ground distance
- D — used 80 cos 25°, which is neither
1.4
In △ABC, ∠B = 90°, AC = 26 and BC = 10. Then AB = …
(1)
A)24✓
B)27,86
C)16
D)36
Answer: A — By Pythagoras AB² = 26² − 10² = 576, so AB = 24.
- B — ADDED the squares instead of subtracting them
- C — subtracted the two sides instead of their squares
- D — added the two given sides
1.5
A ramp rises 1,5 m over a horizontal distance of 8 m. Its angle of inclination is …
(1)
A)10,62°✓
B)10,81°
C)79,38°
D)0,19°
Answer: A — tan θ = 1,5 ÷ 8 = 0,1875, so θ = 10,62°.
- B — used sin⁻¹(1,5 ÷ 8), but 8 m is the horizontal run, not the slope length
- C — inverted the ratio — that is the angle at the top of the ramp
- D — wrote the ratio 0,19 down as though it were the angle
1.6
In △PQR, ∠Q = 90°, ∠P = 32° and PR = 15 cm. Then QR = …
(1)
A)7,95 cm✓
B)12,72 cm
C)9,37 cm
D)28,31 cm
Answer: A — PR is the hypotenuse and QR lies opposite ∠P, so QR = 15 sin 32° = 7,95 cm.
- B — that is 15 cos 32°, which gives PQ, the adjacent side
- C — used 15 tan 32°, but tan never involves the hypotenuse
- D — DIVIDED 15 by sin 32° instead of multiplying
1.7
Two buildings stand 30 m apart. From the top of the shorter one the angle of elevation of the top of the taller one is 40°. The taller building is higher by …
(1)
A)25,17 m✓
B)35,75 m
C)19,28 m
D)39,16 m
Answer: A — The 30 m gap is the adjacent side, so the difference in height is 30 tan 40° = 25,17 m.
- B — DIVIDED 30 by tan 40° instead of multiplying
- C — used 30 sin 40°, but 30 m is the adjacent side, not the hypotenuse
- D — used 30 ÷ cos 40°, which gives the line of sight
1.8
Which ratio finds an ANGLE when the opposite side and the hypotenuse are known?
(1)
A)sin⁻¹✓
B)cos⁻¹
C)tan⁻¹
D)sin
Answer: A — Opposite over hypotenuse is a SINE, so the inverse sine returns the angle.
- B — cosine uses the ADJACENT side with the hypotenuse
- C — tangent uses the opposite with the ADJACENT side, never the hypotenuse
- D — sin turns an angle into a ratio; the inverse is needed to go the other way
1.9
A kite string 50 m long makes an angle of 58° with the ground. The kite is … above the ground.
(1)
A)42,40 m✓
B)26,50 m
C)80,02 m
D)58,96 m
Answer: A — The string is the hypotenuse, so the height is 50 sin 58° = 42,40 m.
- B — that is 50 cos 58°, the horizontal distance to the kite
- C — used 50 tan 58°, but tan never involves the hypotenuse
- D — DIVIDED 50 by sin 58° instead of multiplying
1.10
In △ABC, ∠B = 90°. Then ∠A = 90° − ∠C because …
(1)
A)the three angles of a triangle add up to 180°✓
B)the two acute angles are always equal
C)∠B is the largest angle
D)the sides are always in the ratio 3 : 4 : 5
Answer: A — With ∠B = 90° the other two angles must add to 90°, so each is the complement of the other.
- B — they add to 90° but are equal only in the 45°-45°-90° case
- C — ∠B being the largest is true, but it says nothing about the other two
- D — 3 : 4 : 5 is ONE particular right-angled triangle, not every one

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

[10 MARKS]

In the diagram below, a vertical tower stands on level ground. From a point 25 m from the foot of the tower, the angle of elevation to the top of the tower is 38°. The height of the tower is h.

2.1
Write down which trigonometric ratio relates the 25 m, the 38° and the height h, and give a reason for your choice.
(2)
tan, because the 25 m is ADJACENT to the 38° angle and h is OPPOSITE it (1)
tan θ = opposite ÷ adjacent, and the hypotenuse is neither given nor asked for (1)
2.2
Calculate the height of the tower, correct to TWO decimal places.
(4)
tan 38° = h25 (2)
h = 25 tan 38° (1) = 19,53 m (1)
2.3
The observer now moves to a point where the angle of elevation is 50°. Determine, with a calculation, whether the observer is closer to or further from the tower.
(4)
The height stays 19,53 m (1)
New distance = 19,53tan 50° (2) ≈ 16,39 m, so the observer is CLOSER (1)

Question 3

[10 MARKS]

Answer the questions below. Round off to TWO decimal places where necessary.

3.1
In a right-angled triangle the side opposite a 40° angle is x and the adjacent side is 9. Calculate x.
(3)
tan 40° = x9 (1)
x = 9 tan 40° (1) = 7,55 (1)
3.2
The angle of DEPRESSION from the top of a 40 m cliff to a boat is 22°. Calculate how far the boat is from the foot of the cliff.
(4)
The angle of depression equals the angle of elevation from the boat (1)
tan 22° = 40d (1)
d = 40tan 22° (1) = 99,00 m (1)
3.3
Explain the difference between an angle of elevation and an angle of depression, and why they are equal in the question above.
(3)
Elevation is measured UP from the horizontal, depression DOWN from it (2)
The two horizontals are parallel, so the angles are alternate and therefore equal (1)

TOTAL: 30 marks

This question paper consists of 3 questions.