Coordinates and trigonometry · Ages 14–15

Trigonometry in plane figures 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.A rectangle's diagonal is 37 cm long and makes 16° with a long side (the picture is half of it). Correct to 1 decimal place, what is the long side x, in cm?
Answer:
2.A rectangle's diagonal is 27 cm long and makes 24° with a long side (the picture is half of it). Correct to 1 decimal place, what is the long side x, in cm?
Answer:
3.An isosceles triangle has two equal sides of 22 cm and an angle of 76° at the top. Correct to 1 decimal place, how long is its base, in cm?
Answer:
4.In triangle ABC in the picture the height AD = 7 cm, ∠B = 56° and ∠C = 50°. Correct to 1 decimal place, what is BC, in cm?
Answer:
5.A regular 5-sided polygon has sides of 18 cm. Correct to 1 decimal place, how far is its centre from a side, in cm?
Answer:
6.A rhombus has sides of 17 cm and an acute angle of 44°. Correct to 1 decimal place, how long is its longer diagonal, in cm?
Answer:
7.In triangle ABC in the picture the height AD = 12 cm, ∠B = 61° and ∠C = 68°. Correct to 1 decimal place: x = BD = cm, y = DC = cm.
Fill in each blank
Answer:
8.A rhombus has sides of 10 cm and an acute angle of 52°. Correct to 1 decimal place, how long is its longer diagonal, in cm?
Answer:
9.A regular 8-sided polygon has sides of 11 cm. Correct to 1 decimal place, how far is its centre from a side, in cm?
Answer:

10.In triangle ABC in the picture the height AD = 20 cm, ∠B = 58° and ∠C = 54°. What is BD?
- A. 20 ÷ sin 58°
- B. 20 × tan 58°
- C. 20 × sin 58°
- D. 20 ÷ tan 58°
Choose the right answer
Answer:
11.In triangle ABC in the picture the height AD = 18 cm, ∠B = 46° and ∠C = 43°. Correct to 1 decimal place, AB = 25.9 cm.
True or false?
Answer:
12.In triangle ABC in the picture the height AD = 4 cm, ∠B = 55° and ∠C = 43°. Correct to 1 decimal place, AB = 4.9 cm.
True or false?
Answer:

13.A symmetrical pitched roof has two rafters of 9 m, each at 20° to the horizontal. Correct to 1 decimal place, how far does the roof span, in m?
Working:
Answer:
14.A rhombus-shaped kite has sides of 16 cm and an angle of 58°. Correct to 1 decimal place: the shorter diagonal is cm, the longer cm.
Fill in each blank
Working:
Answer:
15.A table is a regular hexagon with sides of 75 cm. Correct to 1 decimal place, how far apart are two opposite sides, in cm?
Working:
Answer:
16.A slide's frame is 5 m high: on one side the steps, at 49° to the ground; on the other the slide, at 33° to the ground. Correct to 1 decimal place, how far apart are the foot of the steps and the end of the slide, in m?
Working:
Answer:
17.A rhombus-shaped kite has sides of 40 cm and an angle of 72°. Correct to 1 decimal place: the shorter diagonal is cm, the longer cm.
Fill in each blank
Working:
Answer:
18.A table is a regular hexagon with sides of 80 cm. Correct to 1 decimal place, how far apart are two opposite sides, in cm?
Working:
Answer:

Show the answers and working

Trigonometry in plane figures worksheet: Answers and working

1. 35.6 cm
The diagonal, a long side and a short side make a right-angled triangle, the diagonal its hypotenuse: x = 37 cos 16° ≈ 35.6 cm.
2. 24.7 cm
The diagonal, a long side and a short side make a right-angled triangle, the diagonal its hypotenuse: x = 27 cos 24° ≈ 24.7 cm.
3. 27.1 cm
The height from the top halves the top angle into 38° and the base into two: half the base = 22 sin 38°, the base = 2 × 22 sin 38° ≈ 27.1 cm.
4. 10.6 cm
BD = 7 ÷ tan 56°, DC = 7 ÷ tan 50°, BC = BD + DC ≈ 10.6 cm (rounded at the end).
5. 12.4 cm
Join the centre to a side's ends and middle: a right-angled triangle with angle 360° ÷ 5 ÷ 2 = 36° and half a side, 9, opposite it: distance = 9 ÷ tan 36° ≈ 12.4 cm.
6. 31.5 cm
The diagonals bisect each other at right angles, the longer one halving the 44° angle: half of it = 17 cos 22°, the whole 2 × 17 cos 22° ≈ 31.5 cm.
7. 6.7, 4.8
The two right-angled triangles share AD: BD = 12 ÷ tan 61° ≈ 6.7, DC = 12 ÷ tan 68° ≈ 4.8.
8. 18 cm
The diagonals bisect each other at right angles, the longer one halving the 52° angle: half of it = 10 cos 26°, the whole 2 × 10 cos 26° ≈ 18 cm.
9. 13.3 cm
Join the centre to a side's ends and middle: a right-angled triangle with angle 360° ÷ 8 ÷ 2 = 22.5° and half a side, 5.5, opposite it: distance = 5.5 ÷ tan 22.5° ≈ 13.3 cm.

10. D. 20 ÷ tan 58°
In the right-angled triangle ABD, tan 58° = ADBD = 20BD, so BD = 20 ÷ tan 58°.
11. False
AD is ∠B's opposite side and AB the hypotenuse: sin 46° = 18AB, AB = 18 ÷ sin 46° ≈ 25 cm.
12. True
AD is ∠B's opposite side and AB the hypotenuse: sin 55° = 4AB, AB = 4 ÷ sin 55° ≈ 4.9 cm.

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

13. 16.9 m
An upright from the ridge splits it into two right-angled triangles: each spans 9 cos 20°, so 2 × 9 cos 20° ≈ 16.9 m in all.
14. 15.5, 28
The diagonals bisect each other at right angles, halving 58° into 29°: the short one 2 × 16 sin 29° ≈ 15.5, the long 2 × 16 cos 29° ≈ 28.
15. 129.9 cm
A regular hexagon is 6 equilateral triangles; the centre is an equilateral triangle's height, 75 cos 30°, from a side, so opposite sides are 2 × 75 cos 30° ≈ 129.9 cm apart.
16. 12 m
An upright from the top splits it into two right-angled triangles: 5 ÷ tan 33° + 5 ÷ tan 49° ≈ 12 m.
17. 47, 64.7
The diagonals bisect each other at right angles, halving 72° into 36°: the short one 2 × 40 sin 36° ≈ 47, the long 2 × 40 cos 36° ≈ 64.7.
18. 138.6 cm
A regular hexagon is 6 equilateral triangles; the centre is an equilateral triangle's height, 80 cos 30°, from a side, so opposite sides are 2 × 80 cos 30° ≈ 138.6 cm apart.

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