Coordinates and trigonometry · Ages 14–15
How the ratios are related worksheet
Part of: Trigonometry
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How the ratios are related 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.θ is acute and sin θ = 1517. What is cos θ, as a fraction?
Answer in simplest form
Answer:
2.θ is acute and sin θ = 35. What is cos θ, as a fraction?
Answer in simplest form
Answer:
3.Given sin 72° ≈ 0.9511, without a calculator, cos 18° ≈ (4 decimal places)
Answer:
4.θ is acute and cos θ = 1213. What is 1 − sin² θ, as a fraction?
Answer in simplest form
Answer:
5.θ is acute and tan θ = 512. As fractions: sin θ = , cos θ = .
Fill in each blank
Answer:
6.θ is acute and tan θ = 815. As fractions: sin θ = , cos θ = .
Fill in each blank
Answer:
7.Given sin 62° ≈ 0.8829, without a calculator, cos 28° ≈ (4 decimal places)
Answer:
8.θ is acute and cos θ = 817. What is 1 + tan² θ, as a fraction?
Answer in simplest form
Answer:
9.θ is acute and sin θ = 2129. What is tan θ + 1tan θ, as a fraction?
Answer in simplest form
Answer:
Part B: Choose, or true or false (2 marks each)
10.θ is acute, with sin θ = 2129 and cos θ = 2029. What is tan θ?
- A. 2120
- B. 2920
- C. 420841
- D. 2021
Choose the right answer
Answer:
11.There can be an acute angle θ with sin θ = 513 and cos θ = 1213.
True or false?
Answer:
12.θ is acute, with sin θ = 45 and cos θ = 35. What is tan θ?
- A. 34
- B. 43
- C. 53
- D. 1225
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.A ramp's angle θ with the ground has sin θ = 1213. An engineer wants the other two ratios: cos θ = , tan θ = (as fractions).
Fill in each blank
Working:
Answer:
14.A surveyor's table gives cos 73° ≈ 0.2924, but he needs sin 17°. Without a calculator, what is sin 17°, to 4 decimal places?
Working:
Answer:
15.A mountain path's inclination θ has tan θ = 43. Walking 45 m along it, how many metres do you climb?
Working:
Answer:
16.θ is acute and sin θ = 35. From sin² θ + cos² θ = 1, find tan θ: put the steps in order.
- A. cos θ = 45
- B. cos² θ = 1 − (35)2 = 1625
- C. tan θ = sin θ ÷ cos θ = 34
Write the letters in the right order
Working:
Answer:
17.A ramp's angle θ with the ground has sin θ = 513. An engineer wants the other two ratios: cos θ = , tan θ = (as fractions).
Fill in each blank
Working:
Answer:
18.A surveyor's table gives cos 28° ≈ 0.8829, but he needs sin 62°. Without a calculator, what is sin 62°, to 4 decimal places?
Working:
Answer:
Show the answers and working
How the ratios are related worksheet: Answers and working
Part A: Working out and filling in
1. 817
cos² θ = 1 − sin² θ = 1 − (1517)2, so cos θ = 817 (positive for an acute angle). Or draw the triangle: the adjacent side 172 − 152 = 8.
2. 45
cos² θ = 1 − sin² θ = 1 − (35)2, so cos θ = 45 (positive for an acute angle). Or draw the triangle: the adjacent side 52 − 32 = 4.
3. 0.9511
72° + 18° = 90°: cos(90° − θ) = sin θ, so cos 18° = sin 72° ≈ 0.9511.
4. 144169
sin² θ + cos² θ = 1, so 1 − sin² θ = cos² θ = (1213)2 = 144169, with no need for sin θ first.
5. 513, 1213
Draw a right-angled triangle with opposite 5 and adjacent 12: the hypotenuse is 52 + 122 = 13, so sin θ = 513 and cos θ = 1213.
6. 817, 1517
Draw a right-angled triangle with opposite 8 and adjacent 15: the hypotenuse is 82 + 152 = 17, so sin θ = 817 and cos θ = 1517.
7. 0.8829
62° + 28° = 90°: cos(90° − θ) = sin θ, so cos 28° = sin 62° ≈ 0.8829.
8. 28964
1 + tan² θ = (cos² θ + sin² θ)/cos² θ = 1cos² θ = (178)2 = 28964.
9. 841420
cos θ = 2029, tan θ = 2120: 2120 + 2021 = 212 + 20220 × 21 = 841420. (It also equals 1 ÷ (sin θ cos θ).)
Part B: Choose, or true or false
10. A. 2120
tan θ = sin θ ÷ cos θ = 2129 ÷ 2029 = 2120.
11. True
sin² θ + cos² θ must be 1: (513)2 + (1213)2 = 1, so cos θ can only be 1213.
12. B. 43
tan θ = sin θ ÷ cos θ = 45 ÷ 35 = 43.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 513, 125
cos² θ = 1 − (1213)2 = (513)2, so cos θ = 513; tan θ = sin ÷ cos = 125.
14. 0.2924
73° and 17° add to 90°: sin(90° − θ) = cos θ, so sin 17° = cos 73° ≈ 0.2924.
15. 36 m
tan θ = 43, the hypotenuse 42 + 32 = 5, so sin θ = 45; the climb = 45 × 45 = 36 m.
16. B → A → C
cos² θ first, its square root (positive for an acute angle), then tan = sin ÷ cos.
17. 1213, 512
cos² θ = 1 − (513)2 = (1213)2, so cos θ = 1213; tan θ = sin ÷ cos = 512.
18. 0.8829
28° and 62° add to 90°: sin(90° − θ) = cos θ, so sin 62° = cos 28° ≈ 0.8829.
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