Trigonometry

14 Grade 10 Essential Mathematics questions on Trigonometry, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   4 marks

   A rectangular maize plot measures 60 m by 25 m. A farmer clears a straight path along one diagonal of the plot. Calculate the length of the path and the angle it makes with the longer side, to 1 decimal place.

   (4 marks)

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2. Question 2

   1 mark

   The sloping roof of a poultry house makes an angle θ with the horizontal, where tan θ = 0.75. State the size of θ to the nearest degree.

   A. 53°

   B. 37°

   C. 41°

   D. 48°

   (1 mark)

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3. Question 3

   1 mark

   A carpenter in Thika makes a right-angled wooden bracket whose two shorter sides measure 8 cm and 15 cm. Calculate the length of the longest side of the bracket.

   A. 7 cm

   B. 17 cm

   C. 23 cm

   D. 12.7 cm

   (1 mark)

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4. Question 4

   1 mark

   A fisherman standing on a cliff at Mbita looks downwards at a boat on the lake. Identify the name given to the angle between his horizontal line of sight and his line of sight to the boat.

   A. Gradient

   B. Angle of depression

   C. Angle of elevation

   D. Bearing

   (1 mark)

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5. Question 5

   3 marks

   Distinguish between the angle of elevation and the angle of depression, and give one everyday example of each.

   (3 marks)

   Marking scheme

   Angle of elevation: the angle measured upwards from the horizontal line of sight of the observer to an object above — 1 mark. Angle of depression: the angle measured downwards from the horizontal line of sight to an object below — 1 mark. Example of each, e.g. elevation — a pupil on the ground looking up at the top of a school water tank; depression — a driver on a hill above Naivasha looking down at the road below — 1 mark for a sensible example of each.

6. Question 6

   4 marks

   A surveyor stands 40 m from the foot of a communication mast on level ground. The angle of elevation of the top of the mast from her eye is 38°, and her eye is 1.5 m above the ground. Calculate the height of the mast, to 1 decimal place.

   (4 marks)

   Marking scheme

   tan 38° = h/40, where h is the height of the mast above eye level — 1 mark. $h=40\times 0.7813=31.25m$ — 1 mark. The mast also extends from the ground up to eye level, so add 1.5 m — 1 mark. Height of mast = 31.25 + 1.5 = 32.75 m = 32.8 m (1 d.p.) — 1 mark.

7. Question 7

   1 mark

   From the top of a cliff 45 m high, the angle of depression of a boat on the water below is 30°. Calculate the distance of the boat from the foot of the cliff, to 1 decimal place.

   A. 90.0 m

   B. 77.9 m

   C. 26.0 m

   D. 22.5 m

   (1 mark)

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8. Question 8

   1 mark

   A support cable 20 m long runs from the top of a water tank to a peg on level ground and makes an angle of 35° with the ground. Calculate the height of the point where the cable is fixed on the tank, to 1 decimal place.

   A. 34.9 m

   B. 16.4 m

   C. 11.5 m

   D. 14.0 m

   (1 mark)

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9. Question 9

   3 marks

   A ladder 5 m long leans against the wall of a shop with its foot 1.4 m from the base of the wall on level ground. Calculate the angle the ladder makes with the ground, to 1 decimal place.

   (3 marks)

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10. Question 10

    1 mark

    State the value of cos 60°, correct to four decimal places.

    A. 0.5774

    B. 0.8660

    C. 1.7321

    D. 0.5000

    (1 mark)

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11. Question 11

    3 marks

    An angle θ is acute and $\tan \theta =\frac{5}{12}$. State the length of the hypotenuse of a right-angled triangle with these two sides, and hence state sin θ and cos θ.

    (3 marks)

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12. Question 12

    3 marks

    Describe how a learner decides whether to use the sine, the cosine or the tangent ratio when working with a right-angled triangle.

    (3 marks)

    Marking scheme

    First label the three sides in relation to the angle being used: the hypotenuse, the side opposite the angle and the side adjacent to the angle — 1 mark. If the two sides involved (given plus required) are the opposite and the hypotenuse, use sine; if they are the adjacent and the hypotenuse, use cosine — 1 mark. If they are the opposite and the adjacent, use tangent — 1 mark.

13. Question 13

    3 marks

    The sloping rafter of a roof at a school in Machakos is 4.5 m long and makes an angle of 28° with the horizontal beam on which it rests. Calculate the horizontal distance covered by the rafter and the vertical rise from the beam to the upper end of the rafter, each to 2 decimal places.

    (3 marks)

    Marking scheme

    With the 4.5 m rafter as the hypotenuse: cos 28° = b/4.5 for the horizontal distance b, and sin 28° = h/4.5 for the vertical rise h — 1 mark. $b=4.5\times 0.8829=3.97m$ (2 d.p.) — 1 mark. $h=4.5\times 0.4695=2.11m$ (2 d.p.) — 1 mark.

14. Question 14

    1 mark

    A flagpole at a county stadium stands 12 m tall and casts a shadow 5 m long on level ground. State the tangent of the angle of elevation of the sun.

    A. 0.4167

    B. 0.3846

    C. 0.9231

    D. 2.4000

    (1 mark)

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