Coordinates and trigonometry · Ages 14–15

Sine, cosine and tangent 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 sides of 45 cm, 24 cm and 51 cm. What is tan θ, as a fraction in its lowest terms?
Answer in simplest form
Answer:
2.Find cos 38° on a calculator, correct to 4 decimal places: cos 38° ≈
Answer:
3.The right-angled triangle in the picture has a hypotenuse of 75 cm, and θ's opposite side is 21 cm. What is cos θ, as a fraction in its lowest terms?
Answer in simplest form
Answer:
4.The right-angled triangle in the picture has sides of 20 cm, 21 cm and 29 cm. As fractions: sin θ = , cos θ = , tan θ = .
Fill in each blank
Answer:
5.The right-angled triangle in the picture has a hypotenuse of 50 cm, and θ's opposite side is 14 cm. What is cos θ, as a fraction in its lowest terms?
Answer in simplest form
Answer:
6.In a right-angled triangle an acute angle θ has tan θ = 2021, and the hypotenuse is 145 cm. How long is the side opposite θ, in cm?
Answer:
7.θ and φ are the two acute angles of a right-angled triangle, and sin θ = 817. What is tan φ, as a fraction in its lowest terms?
Answer in simplest form
Answer:
8.θ and φ are the two acute angles of a right-angled triangle, and sin θ = 2129. What is tan φ, as a fraction in its lowest terms?
Answer in simplest form
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9.In a right-angled triangle an acute angle θ has tan θ = 34, and the hypotenuse is 20 cm. How long is the side opposite θ, in cm?
Answer:

10.The right-angled triangle in the picture has sides of 20 cm, 21 cm and 29 cm. What is sin θ?
- A. 2029
- B. 2129
- C. 2021
- D. 2120
Choose the right answer
Answer:
11.The right-angled triangle in the picture has sides of 4 cm, 3 cm and 5 cm. What is sin θ?
- A. 35
- B. 43
- C. 45
- D. 34
Choose the right answer
Answer:
12.The right-angled triangle in the picture has sides of 20 cm, 21 cm and 29 cm, and tan θ = 2120.
True or false?
Answer:

13.A right-angled triangle has a hypotenuse of 5 cm, and θ's opposite side is 3 cm. Find tan θ: put the steps in order.
- A. tan θ = 34
- B. Adjacent² = 5² − 3² = 16
- C. Adjacent = 4
Write the letters in the right order
Working:
Answer:
14.A cable car's line makes 7° with the horizontal, so every 1 m along it rises sin 7° m. With a calculator, correct to 4 decimal places, what is sin 7°?
Working:
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15.A slope has cos θ = 1213, θ its angle with the ground. Walking 78 m up it, how far do you move across, in m?
Working:
Answer:
16.A ladder 13 m long leans on a wall, its foot 12 m from the wall, at an angle θ to the ground. Its top is m up; cos θ = and tan θ = (as fractions).
Fill in each blank
Working:
Answer:
17.A slope has cos θ = 513, θ its angle with the ground. Walking 104 m up it, how far do you move across, in m?
Working:
Answer:
18.A ladder 25 m long leans on a wall, its foot 24 m from the wall, at an angle θ to the ground. Its top is m up; cos θ = and tan θ = (as fractions).
Fill in each blank
Working:
Answer:

Show the answers and working

Sine, cosine and tangent worksheet: Answers and working

1. 815
θ's opposite side is 24, its adjacent 45: tan θ = 2445 = 815.
2. 0.788
With the calculator in degrees (DEG), key cos 38 and round to 4 decimal places: 0.788.
3. 2425
The adjacent side = 75² − 21² = 72 (Pythagoras); cos θ = 7275 = 2425.
4. 2029, 2129, 2021
Find θ's opposite and adjacent sides and the hypotenuse 29 in the picture: sin = opp ÷ hyp = 2029, cos = adj ÷ hyp = 2129, tan = opp ÷ adj = 2021.
5. 2425
The adjacent side = 50² − 14² = 48 (Pythagoras); cos θ = 4850 = 2425.
6. 100 cm
Opposite : adjacent = 20 : 21, so the hypotenuse goes with 20² + 21² = 29; 145 ÷ 29 = 5 times, so the opposite side is 20 × 5 = 100 cm.
7. 158
Draw it: θ's opposite side 8, hypotenuse 17, adjacent 17² − 8² = 15. φ's opposite side is θ's adjacent: tan φ = 158.
8. 2021
Draw it: θ's opposite side 21, hypotenuse 29, adjacent 29² − 21² = 20. φ's opposite side is θ's adjacent: tan φ = 2021.
9. 12 cm
Opposite : adjacent = 3 : 4, so the hypotenuse goes with 3² + 4² = 5; 20 ÷ 5 = 4 times, so the opposite side is 3 × 4 = 12 cm.

10. B. 2129
The hypotenuse is 29 (the longest, facing the right angle); sin θ = θ's opposite side ÷ 29.
11. C. 45
The hypotenuse is 5 (the longest, facing the right angle); sin θ = θ's opposite side ÷ 5.
12. True
tan θ = θ's opposite ÷ adjacent = 2120, not upside down, 2021.

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

13. B → C → A
tan needs the opposite and adjacent sides: Pythagoras gives the adjacent 4 first, then tan θ = 34.
14. 0.1219 m
In degrees: sin 7° ≈ 0.1219.
15. 72 m
cos θ = across ÷ along: across = 78 × 1213 = 72 m.
16. 5, 1213, 512
Pythagoras: 13² − 12² = 5; cos θ = adj ÷ hyp = 1213, tan θ = opp ÷ adj = 512.
17. 40 m
cos θ = across ÷ along: across = 104 × 513 = 40 m.
18. 7, 2425, 724
Pythagoras: 25² − 24² = 7; cos θ = adj ÷ hyp = 2425, tan θ = opp ÷ adj = 724.

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