Coordinates and trigonometry · Ages 14–15

Exact values at 30°, 45° and 60° 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.Without a calculator: 5 tan 45° − 7 cos 60° =
Answer:
2.Without a calculator: 8 sin 30° + 3 tan 45° =
Answer:
3.Without a calculator: 3 sin 30° + 7 tan 45° =
Answer:
4.The right-angled triangle in the picture has a hypotenuse of 8 cm, and the angle marked is 30°. In exact values: x = cm, y = cm.
Fill in each blank
Answer:
5.Without a calculator: 8 sin² 60° + 3 cos² 45° =
Answer:
6.Without a calculator: 6 sin² 60° + 8 cos² 45° =
Answer:
7.In a right-angled triangle the side beside a 30° angle is 27 cm and the side opposite it is x cm. In exact values: x =
Answer:
8.A right-angled isosceles triangle has a hypotenuse of 142 cm. How long is each shorter side, in cm?
Answer:
9.A right-angled isosceles triangle has a hypotenuse of 102 cm. How long is each shorter side, in cm?
Answer:
10.In a right-angled triangle the side beside a 30° angle is 30 cm and the side opposite it is x cm. In exact values: x =
Answer:

13.A ladder 18 m long leans on a wall at 60° to the ground. In exact values: its foot is m from the wall, and its top m up.
Fill in each blank
Working:
Answer:
14.A road sign is an equilateral triangle with sides of 8 dm. In exact values, its height is dm
Working:
Answer:
15.From the ground 373 m from a tower, the angle of elevation of its top is 30°. Without a calculator, how tall is the tower, in m?
Working:
Answer:
16.A symmetrical roof spans 24 m, each side at 45° to the horizontal. In exact values, each side runs m from the eave to the ridge
Working:
Answer:
17.A road sign is an equilateral triangle with sides of 24 dm. In exact values, its height is dm
Working:
Answer:
18.A symmetrical roof spans 2 m, each side at 45° to the horizontal. In exact values, each side runs m from the eave to the ridge
Working:
Answer:

Show the answers and working

Exact values at 30°, 45° and 60° worksheet: Answers and working

11. A. 22
sin 30° = 12, sin 45° = 22, sin 60° = 32: from half an equilateral triangle and a right-angled isosceles one.
12. False
sin θ = cos(90° − θ): sin 30° equals cos(90° − 30°). Check with exact values: sin 30° = cos 60° = 12, sin 45° = cos 45° = 22.

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

13. 9, 243
18 cos 60° = 18 × 12 = 9; 18 sin 60° = 18 × 32 = 93 = 243.
14. 48
Height = 8 × sin 60° = 8 × 32 = 43 = 48 dm. Or by Pythagoras: 8² − 4².
15. 37 m
Height = 373 × tan 30° = 373 × 13 = 37 m.
16. 288
Each side covers 12 m across and the roof is the hypotenuse: 12 ÷ cos 45° = 122 = 288 m.
17. 432
Height = 24 × sin 60° = 24 × 32 = 123 = 432 dm. Or by Pythagoras: 24² − 12².
18. 2
Each side covers 1 m across and the roof is the hypotenuse: 1 ÷ cos 45° = 12 = 2 m.

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