Coordinates and trigonometry · Ages 13–14

How the ratios change with the angle worksheet

Part of: Trigonometry (I)

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Coordinates and trigonometry · Ages 13–14

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.With a calculator, correct to 4 decimal places: sin 18° + sin 16° ≈ and sin 34° ≈ .
Fill in each blank
Answer:
2.θ is an acute angle with cos θ = 13n, n a positive whole number no more than 33. How many different values can n take?
Answer:
3.θ is an acute angle with sin θ = n4, n a positive whole number. How many different values can n take?
Answer:

4.θ is an acute angle. As θ grows from 36° to 81°, what happens to cos θ?
- A. It gets bigger
- B. It stays the same
- C. It gets smaller
- D. Up first, then down
Choose the right answer
Answer:
5.There can be an acute angle θ with cos θ = 1.4.
True or false?
Answer:
6.No calculator needed: sin 85° is bigger than sin 46°.
True or false?
Answer:
7.No calculator needed: sin 34° is bigger than sin 74°.
True or false?
Answer:
8.θ is an acute angle with tan θ = 0.2, so θ is more than 45°.
True or false?
Answer:
9.Without a calculator, which of these is the biggest?
- A. cos 81°
- B. cos 29°
- C. cos 11°
- D. cos 53°
Choose the right answer
Answer:
10.θ is an acute angle with tan θ = 3, so θ is more than 45°.
True or false?
Answer:
11.Without a calculator: for θ = 52°, how do sin θ, cos θ and tan θ go, biggest first?
- A. tan θ > sin θ > cos θ
- B. sin θ > cos θ > tan θ
- C. sin θ > tan θ > cos θ
- D. cos θ > sin θ > tan θ
Choose the right answer
Answer:
12.Without a calculator: for θ = 65°, how do sin θ, cos θ and tan θ go, biggest first?
- A. sin θ > tan θ > cos θ
- B. cos θ > sin θ > tan θ
- C. sin θ > cos θ > tan θ
- D. tan θ > sin θ > cos θ
Choose the right answer
Answer:

13.Two slides are both 4 m long: A makes 16° with the ground, B 39°. The top of B is higher than the top of A.
True or false?
Working:
Answer:
14.A slide 7 m long makes 26° with the ground. It is refitted to make 60° (the same length). Correct to 1 decimal place, how much higher is its top?
Working:
Answer:
15.A ladder 3 m long leans on a wall. Its foot is moved closer, and its angle with the ground grows from 49° to 77°. What happens to the top's height (3 × the sine of the angle)?
- A. It becomes exactly 7749 times as much
- B. It stays the same
- C. It rises
- D. It falls
Choose the right answer
Working:
Answer:
16.Flying a kite, the string is 70 m long and pulled straight. At 28° to the horizontal the kite is m above the hand; with the same string at double the angle, 56°, it is m above (correct to 1 decimal place).
Fill in each blank
Working:
Answer:
17.Four ramps are each 4 m long, at 13°, 33°, 64° and 76° to the ground. Walking each to the top, which covers the least ground across?
- A. The 64° one
- B. The 33° one
- C. The 13° one
- D. The 76° one
Choose the right answer
Working:
Answer:
18.A ladder 7 m long leans on a wall. Its foot is moved closer, and its angle with the ground grows from 46° to 80°. What happens to the top's height (7 × the sine of the angle)?
- A. It falls
- B. It becomes exactly 8046 times as much
- C. It stays the same
- D. It rises
Choose the right answer
Working:
Answer:

Show the answers and working

How the ratios change with the angle worksheet: Answers and working

1. 0.5847, 0.5592
sin 18° + sin 16° ≈ 0.5847 and sin 34° ≈ 0.5592: not the same. A ratio can't be split up: sin(18° + 16°) ≠ sin 18° + sin 16°.
2. 20
An acute angle's cosine is between 0 and 1: 13n < 1, so n must be over 13; it can be 13 + 1 up to 33, 33 − 13 values.
3. 3
An acute angle's sine is always more than 0 and less than 1 (1 would make θ 90°): 0 < n4 < 1, so n can be 1 up to 4 − 1.

4. C. It gets smaller
Fix the hypotenuse and let θ grow from 36° to 81°: the adjacent side gets shorter and shorter, so cos θ = adjacent ÷ hypotenuse keeps falling.
5. False
cos θ = adjacent ÷ hypotenuse, and the hypotenuse is the longest side, so an acute angle's cosine is always between 0 and 1: 1.4 must be under 1.
6. True
The bigger the acute angle, the longer its opposite side and the bigger its sine: compare 85° with 46°.
7. False
The bigger the acute angle, the longer its opposite side and the bigger its sine: compare 34° with 74°.
8. False
tan θ = opposite ÷ adjacent. At 45° the two are equal and tan θ = 1; the bigger the angle, the bigger its tangent, so a tangent over 1 means θ over 45°, under 1 means under 45°.
9. C. cos 11°
The bigger an acute angle, the smaller its cosine, so cos 11° > cos 29° > cos 53° > cos 81°.
10. True
tan θ = opposite ÷ adjacent. At 45° the two are equal and tan θ = 1; the bigger the angle, the bigger its tangent, so a tangent over 1 means θ over 45°, under 1 means under 45°.
11. A. tan θ > sin θ > cos θ
52° is more than 45°: the opposite side is longer than the adjacent, so sin θ > cos θ, and tan θ = opp ÷ adj is over 1 while sin θ and cos θ are under 1. So tan θ > sin θ > cos θ.
12. D. tan θ > sin θ > cos θ
65° is more than 45°: the opposite side is longer than the adjacent, so sin θ > cos θ, and tan θ = opp ÷ adj is over 1 while sin θ and cos θ are under 1. So tan θ > sin θ > cos θ.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. True
A top's height = 4 × the sine of its angle. With the same length, the bigger the angle, the bigger its sine and the higher the top: compare 16° with 39°.
14. 3 m
It rises 7 sin 60° − 7 sin 26° ≈ 3 m. 7 sin 34° won't do: sin 60° − sin 26° isn't sin(60° − 26°).
15. C. It rises
The top's height = 3 × the sine of the angle. The angle grows from 49° to 77°, so its sine grows and the top rises.
16. 32.9, 58
Height = string × the sine of the angle: 70 sin 28° ≈ 32.9 and 70 sin 56° ≈ 58. Doubling the angle doesn't double the height: sin 56° isn't 2 sin 28°.
17. D. The 76° one
The distance across is the adjacent side: 4 × the cosine of the angle. The bigger the angle, the smaller its cosine, so the 76° ramp covers least ground.
18. D. It rises
The top's height = 7 × the sine of the angle. The angle grows from 46° to 80°, so its sine grows and the top rises.

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