Coordinates and trigonometry · Ages 14–15
Trigonometry worksheet
Covers: Sine, cosine and tangent, Finding a side, Finding an angle, How the ratios are related, Exact values at 30°, 45° and 60°, Trigonometry in plane figures, Gradient and inclination, Angles of elevation and depression, Bearings
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Trigonometry worksheet
Coordinates and trigonometry · Ages 14–15
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.The right-angled triangle in the picture has sides of 3 cm, 4 cm and 5 cm. As fractions: sin θ = , cos θ = , tan θ = .
Fill in each blank
Answer:
2.θ and φ are the two acute angles of a right-angled triangle, and sin θ = 2029. What is tan φ, as a fraction in its lowest terms?
Answer in simplest form
Answer:
3.In the right-angled triangle in the picture the angle marked is 59°, and the side opposite it is 11 cm. Correct to 1 decimal place, what is x, in cm?
Answer:
4.A rectangle is 21 cm long and 10 cm wide. To the nearest 0.1°, what angle does its diagonal make with a long side?
Answer:
5.θ is acute and cos θ = 513. What is 1 − sin² θ, as a fraction?
Answer in simplest form
Answer:
6.A slope has a gradient of 1 : 11. To the nearest 0.1°, what is its inclination?
Answer:
7.From point A on the ground the top of a tower is at an elevation of 25°; walking 31 m nearer, to B, the elevation is 66°. Correct to 1 decimal place, how tall is the tower, in m?
Answer:
8.A ship sails 21 km from O on a bearing of 051° to A, then 20 km on 141° to B. How far is B from O, in km?
Answer:
Part B: Choose, or true or false (2 marks each)
9.The right-angled triangle in the picture has a hypotenuse of 32 cm, and the angle marked is 72°. Correct to 1 decimal place, x = 30.4 cm.
True or false?
Answer:
10.To find the angle marked '?' from the two sides given in the picture, which do you use?
- A. tan⁻¹
- B. cos⁻¹
- C. sin⁻¹
- D. Pythagoras' theorem
Choose the right answer
Answer:
11.What is the exact value of sin 60°?
- A. 12
- B. 3
- C. 32
- D. 22
Choose the right answer
Answer:
12.In triangle ABC in the picture the height AD = 18 cm, ∠B = 44° and ∠C = 60°. Correct to 1 decimal place, AB = 25.9 cm.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.A cable car's line makes 76° with the horizontal, so every 1 m along it rises sin 76° m. With a calculator, correct to 4 decimal places, what is sin 76°?
Working:
Answer:
14.A flagpole 14 m tall casts a shadow 19 m long in the sun. Find the sun's angle of elevation (to the nearest 0.1°): put the steps in order.
- A. θ = tan⁻¹(14 ÷ 19)
- B. θ = 36.4° (to the nearest 0.1°)
- C. tan θ = 14 ÷ 19
Write the letters in the right order
Working:
Answer:
15.A ramp's angle θ with the ground has sin θ = 2029. An engineer wants the other two ratios: cos θ = , tan θ = (as fractions).
Fill in each blank
Working:
Answer:
16.A symmetrical pitched roof has two rafters of 9 m, each at 21° to the horizontal. Correct to 1 decimal place, how far does the roof span, in m?
Working:
Answer:
17.On a map the foot of a hill is 1100 m across from its top, and the top is 110 m higher. To the nearest 0.1°, what is the average inclination?
Working:
Answer:
18.A survey team sees a peak at an elevation of 31° from A, then walks 47 m nearer to B, where the elevation is 54°. Correct to 1 decimal place: the peak is m high, and B is m from the point under it.
Fill in each blank
Working:
Answer:
Show the answers and working
Trigonometry worksheet: Answers and working
Part A: Working out and filling in
1. 45, 35, 43
Find θ's opposite and adjacent sides and the hypotenuse 5 in the picture: sin = opp ÷ hyp = 45, cos = adj ÷ hyp = 35, tan = opp ÷ adj = 43.
2. 2120
Draw it: θ's opposite side 20, hypotenuse 29, adjacent 292 − 202 = 21. φ's opposite side is θ's adjacent: tan φ = 2120.
3. 6.6 cm
x is the adjacent side: tan 59° = 11x, so x = 11 ÷ tan 59° ≈ 6.6 cm. Or by the other acute angle: x = 11 × tan(90° − 59°).
4. 25.5°
The diagonal splits the rectangle into right-angled triangles: tan θ = 1021, θ ≈ 25.5°.
5. 25169
sin² θ + cos² θ = 1, so 1 − sin² θ = cos² θ = (513)2 = 25169, with no need for sin θ first.
6. 5.2°
tan θ = 111, so θ = tan⁻¹(111) ≈ 5.2°.
7. 18.2 m
Let the height be h: A is h ÷ tan 25° from the foot and B h ÷ tan 66°, 31 apart: h(1tan 25° − 1tan 66°) = 31, so h ≈ 18.2 m.
8. 29 km
The two legs' bearings differ by 141° − 51° = 90°, a right angle at A: OB = 212 + 202 = 29 km.
Part B: Choose, or true or false
9. True
x is the opposite side: 32 × sin 72° ≈ 30.4 cm; cos gives the adjacent side.
10. C. sin⁻¹
See whether the two sides are opposite, adjacent or the hypotenuse for the angle '?', and choose the ratio with them.
11. C. 32
sin 30° = 12, sin 45° = 22, sin 60° = 32: from half an equilateral triangle and a right-angled isosceles one.
12. True
AD is ∠B's opposite side and AB the hypotenuse: sin 44° = 18AB, AB = 18 ÷ sin 44° ≈ 25.9 cm.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 0.9703 m
In degrees: sin 76° ≈ 0.9703.
14. C → A → B
The pole is opposite and the shadow adjacent: tan θ = 1419, θ ≈ 36.4°.
15. 2129, 2021
cos² θ = 1 − (2029)2 = (2129)2, so cos θ = 2129; tan θ = sin ÷ cos = 2021.
16. 16.8 m
An upright from the ridge splits it into two right-angled triangles: each spans 9 cos 21°, so 2 × 9 cos 21° ≈ 16.8 m in all.
17. 5.7°
tan θ = 1101100, θ ≈ 5.7°.
18. 50.1, 36.4
Let the height be h: h ÷ tan 31° − h ÷ tan 54° = 47, h ≈ 50.1; B is h ÷ tan 54° ≈ 36.4 from the foot (with h unrounded).
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