Coordinates and trigonometry · Ages 14–15

Finding an angle worksheet

Part of: Trigonometry

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Coordinates and trigonometry · Ages 14–15

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

1.The right-angled triangle in the picture has a hypotenuse of 30 cm, and θ's opposite side is 20 cm. To the nearest 0.1°, what is θ?
Answer:
2.The right-angled triangle in the picture has shorter sides of 11 cm and 14 cm. To the nearest 0.1°, what is θ?
Answer:
3.The right-angled triangle in the picture has a hypotenuse of 15 cm, and θ's opposite side is 11 cm. To the nearest 0.1°, what is θ?
Answer:
4.The right-angled triangle in the picture has a hypotenuse of 30 cm, and the shorter side beside θ is 12 cm. To the nearest 0.1°, what is θ?
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5.The right-angled triangle in the picture has shorter sides of 4 cm and 5 cm. To the nearest 0.1°, what is θ?
Answer:
6.The right-angled triangle in the picture has shorter sides of 16 cm and 18 cm. To the nearest 0.1°, what is θ?
Answer:
7.A rectangle is 20 cm long and 9 cm wide. To the nearest 0.1°, what angle does its diagonal make with a long side?
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8.An isosceles triangle has two equal sides of 12 cm and a base of 15 cm. To the nearest 0.1°, what is each base angle?
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9.An isosceles triangle has two equal sides of 5 cm and a base of 8 cm. To the nearest 0.1°, what is each base angle?
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10.To find the angle marked '?' from the two sides given in the picture, which do you use?
- A. tan⁻¹
- B. Pythagoras' theorem
- C. cos⁻¹
- D. sin⁻¹
Choose the right answer
Answer:
11.The right-angled triangle in the picture has shorter sides of 10 cm and 3 cm. To the nearest 0.1°, θ = 73.3°.
True or false?
Answer:
12.The right-angled triangle in the picture has shorter sides of 3 cm and 12 cm. To the nearest 0.1°, θ = 76°.
True or false?
Answer:

13.A ladder 8 m long leans on a wall, its foot 2 m from the wall. To the nearest 0.1°, what angle does it make with the ground?
Working:
Answer:
14.A rectangular pitch is 110 m long and 50 m wide. Ming runs along a diagonal from one corner to the far one. To the nearest 0.1°, what angle does his path make with a long side?
Working:
Answer:
15.A wheelchair ramp is 3 m long along its surface and rises 20 cm. To the nearest 0.1°, what is its inclination?
Working:
Answer:
16.A flagpole 13 m tall casts a shadow 8 m long in the sun. Find the sun's angle of elevation (to the nearest 0.1°): put the steps in order.
- A. tan θ = 13 ÷ 8
- B. θ = 58.4° (to the nearest 0.1°)
- C. θ = tan⁻¹(13 ÷ 8)
Write the letters in the right order
Working:
Answer:
17.A ladder 7 m long leans on a wall, its foot 3 m from the wall. To the nearest 0.1°, what angle does it make with the ground?
Working:
Answer:
18.A flagpole 6 m tall casts a shadow 14 m long in the sun. Find the sun's angle of elevation (to the nearest 0.1°): put the steps in order.
- A. tan θ = 6 ÷ 14
- B. θ = 23.2° (to the nearest 0.1°)
- C. θ = tan⁻¹(6 ÷ 14)
Write the letters in the right order
Working:
Answer:

Show the answers and working

Finding an angle worksheet: Answers and working

1. 41.8°
sin θ = opposite ÷ hypotenuse = 2030, θ = sin⁻¹(2030) ≈ 41.8°.
2. 51.8°
θ's opposite side is 14 and adjacent 11: tan θ = 1411, θ = tan⁻¹(1411) ≈ 51.8°.
3. 47.2°
sin θ = opposite ÷ hypotenuse = 1115, θ = sin⁻¹(1115) ≈ 47.2°.
4. 66.4°
cos θ = adjacent ÷ hypotenuse = 1230, θ = cos⁻¹(1230) ≈ 66.4°.
5. 38.7°
This θ is at the other corner: opposite 4, adjacent 5, so tan θ = 45 and θ ≈ 38.7°.
6. 41.6°
This θ is at the other corner: opposite 16, adjacent 18, so tan θ = 1618 and θ ≈ 41.6°.
7. 24.2°
The diagonal splits the rectangle into right-angled triangles: tan θ = 920, θ ≈ 24.2°.
8. 51.3°
The height from the top halves the base into 7.5 cm each: cos(base angle) = 7.512, so the base angle ≈ 51.3°.
9. 36.9°
The height from the top halves the base into 4 cm each: cos(base angle) = 45, so the base angle ≈ 36.9°.

10. C. cos⁻¹
See whether the two sides are opposite, adjacent or the hypotenuse for the angle '?', and choose the ratio with them.
11. False
tan θ = opp ÷ adj = 310, θ ≈ 16.7°; upside down gives the other acute angle.
12. True
tan θ = opp ÷ adj = 123, θ ≈ 76°; upside down gives the other acute angle.

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

13. 75.5°
The foot to the wall is adjacent, the ladder the hypotenuse: cos θ = 28, θ ≈ 75.5°.
14. 24.4°
tan θ = width ÷ length = 50110, θ ≈ 24.4°.
15. 3.8°
Same units first: 20 cm = 20100 m; sin θ = rise ÷ length along, θ ≈ 3.8°.
16. A → C → B
The pole is opposite and the shadow adjacent: tan θ = 138, θ ≈ 58.4°.
17. 64.6°
The foot to the wall is adjacent, the ladder the hypotenuse: cos θ = 37, θ ≈ 64.6°.
18. A → C → B
The pole is opposite and the shadow adjacent: tan θ = 614, θ ≈ 23.2°.

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