Trigonometry 1

14 Grade 10 Core Mathematics questions on Trigonometry 1, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   4 marks

   A steel cable runs from the top of a vertical mobile phone mast 25 m high to an anchor peg on level ground, making an angle of 58° with the ground. Calculate (a) the length of the cable, and (b) the distance of the peg from the foot of the mast. Give answers to one decimal place.

   (4 marks)

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2. Question 2

   2 marks

   Explain why the tangent of an acute angle may be greater than 1, while the sine of the same angle can never be greater than 1.

   (2 marks)

   Marking scheme

   tan θ = opposite/adjacent, and in a right-angled triangle the opposite side may be longer than the adjacent side, so the ratio can exceed 1 ✓1 sin θ = opposite/hypotenuse, and the hypotenuse is always the longest side of a right-angled triangle, so the numerator is always smaller than the denominator and the ratio is always less than 1 ✓1

3. Question 3

   1 mark

   In a right-angled triangle, the side opposite an acute angle measures 7 cm and the hypotenuse measures 12 cm. What is the size of that angle, correct to one decimal place?

   A. 30.3°

   B. 54.3°

   C. 35.7°

   D. 59.7°

   (1 mark)

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4. Question 4

   2 marks

   State the exact value of cos 30° and explain why sin 60° has the same value.

   (2 marks)

   Marking scheme

   cos 30° = $\frac{\sqrt{3}}{2}$ ✓1 30° and 60° are complementary (they add up to 90°), and the sine of an angle equals the cosine of its complement, so sin 60° = cos 30° ✓1

5. Question 5

   1 mark

   A learner stands 30 m from the foot of a water tank at her home in Machakos. The angle of elevation of the top of the tank from her eyes is 40°. Which expression gives the height of the top of the tank above the level of her eyes?

   A. 30 sin 40° m

   B. 30 tan 40° m

   C. $30÷\tan 40$° m

   D. 30 cos 40° m

   (1 mark)

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6. Question 6

   1 mark

   For an acute angle θ, which of these expressions is equal to tan θ?

   A. $\cos \theta ÷\sin \theta$

   B. $\sin \theta ÷\cos \theta$

   C. $\sin \theta \times \cos \theta$

   D. sin θ + cos θ

   (1 mark)

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7. Question 7

   1 mark

   A 20 m rope runs from the top of a maize-store pole to a peg on level ground and makes an angle of 30° with the ground. What is the height of the pole?

   A. 40.0 m

   B. 11.5 m

   C. 17.3 m

   D. 10.0 m

   (1 mark)

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8. Question 8

   1 mark

   State the exact value of tan 60° without using a calculator or mathematical tables.

   A. $\frac{1}{\sqrt{3}}$

   B. $\frac{\sqrt{3}}{2}$

   C. $\frac{1}{2}$

   D. $\sqrt{3}$

   (1 mark)

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9. Question 9

   5 marks

   A surveyor on a level plain near Naivasha observes the top of a wind turbine. From point P the angle of elevation is 32°, and from point Q, which is 60 m nearer the turbine and in line with P and its base, the angle of elevation is 48°. Calculate the height of the turbine, correct to one decimal place.

   (5 marks)

   Marking scheme

   Let h be the height. Distance of P from base = $h÷\tan 32$° ✓1 Distance of Q from base = $h÷\tan 48$° ✓1 h/tan 32° − h/tan 48° = 60 ✓1 h(1.6003 − 0.9004) = 60, i.e. $0.6999h=60$ ✓1 h = 85.7 m ✓1

10. Question 10

    3 marks

    For an acute angle θ, $\tan \theta =\frac{5}{12}$. Determine the value of sin θ + cos θ, giving your answer as a fraction in its simplest form.

    (3 marks)

    Marking scheme

    Opposite = 5, adjacent = 12, so hypotenuse = $\sqrt{5^2+12^2}=13$ ✓1 $\sin \theta =\frac{5}{13}$ and $\cos \theta =\frac{12}{13}$ ✓1 $\sin \theta +\cos \theta =\frac{17}{13}$ (= 1 $\frac{4}{13}$ or 1.31) ✓1

11. Question 11

    1 mark

    Identify the true statement about the sine and the cosine of an acute angle as that angle increases from 0° towards 90°.

    A. Both the sine and the cosine increase

    B. The sine decreases while the cosine increases

    C. The sine increases while the cosine decreases

    D. Both the sine and the tangent decrease

    (1 mark)

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12. Question 12

    3 marks

    From the top of a cliff 45 m high along the Kilifi coast, the angle of depression of a fishing boat at sea is 22°. Calculate the horizontal distance of the boat from the foot of the cliff, correct to one decimal place.

    (3 marks)

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13. Question 13

    1 mark

    In a right-angled triangle, the hypotenuse is 13 cm and one of the shorter sides is 5 cm. What is the sine of the angle opposite the 5 cm side?

    A. $\frac{13}{5}$

    B. $\frac{5}{12}$

    C. $\frac{5}{13}$

    D. $\frac{12}{13}$

    (1 mark)

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14. Question 14

    4 marks

    A ladder 7.5 m long rests against a vertical wall with its foot 2.1 m from the base of the wall on level ground. Calculate (a) the angle the ladder makes with the ground, and (b) the height up the wall reached by the ladder. Give answers to one decimal place.

    (4 marks)

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