Printable · GCSE Foundation · ages 14-16
Exact trigonometric values worksheet — GCSE Foundation
Fifteen questions on "exact trigonometric values" — DfE statement G21. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Non-calculator
Answer key: Exact trigonometric values worksheet — GCSE Foundation
MathsUKwww.geekhero.co.uk
- (c) 100√3 cm — The height is opposite the 60° angle, so height = 2 × sin 60° = 2 × √3/2 = √3 m. Converting to centimetres: √3 m = 100√3 cm. '√3 cm' forgets to convert the answer from metres to centimetres. '200√3 cm' comes from mis-recalling sin 60° as √3 instead of √3/2, dropping the denominator of the exact value: 2 × √3 = 2√3 m = 200√3 cm. '50√3 cm' comes from halving the ladder's length before multiplying by sin 60°, instead of using the full 2 m.
- (d) sin 30°, tan 30°, cos 30° — sin 30° = 1/2 = 0.5, tan 30° = √3/3 ≈ 0.577 and cos 30° = √3/2 ≈ 0.866, so the correct order from smallest to largest is sin 30°, tan 30°, cos 30°. 'sin 30°, cos 30°, tan 30°' swaps the last two, wrongly putting cos 30° before tan 30°. 'cos 30°, tan 30°, sin 30°' is the correct list written backwards, from largest to smallest. 'tan 30°, sin 30°, cos 30°' wrongly swaps sin 30° and tan 30° at the start.
- (b) √2/2 — sin 45° = √2/2 (the same value as 1/√2, written with a rational denominator) — one of the exact values you need to know. √3/2 is the exact value of sin 60° and of cos 30°, not sin 45°. 1/2 is the exact value of sin 30° and of cos 60°. 1 is the exact value of sin 90°.
- (d) √3 — tan 60° = √3 — one of the exact values you need to know without a calculator. √3/2 is the exact value of sin 60° and of cos 30°, not tan 60°. 1/√3 is the exact value of tan 30°, the reciprocal-angle case. 3 comes from squaring √3 instead of reading off tan 60° itself.
- (a) 1/2 — cos 0° = 1 and sin 30° = 1/2, so cos 0° − sin 30° = 1 − 1/2 = 1/2. 1 comes from writing down cos 0° alone and forgetting to subtract sin 30°. 3/2 comes from adding the two values instead of subtracting. −1/2 comes from working out sin 30° − cos 0°, the two terms the wrong way round.
- (b) 6√3 m — The horizontal distance covered by one support is adjacent to the 30° angle, so it equals 6 × cos 30° = 6 × √3/2 = 3√3 m. The total base width is made up of both supports, so it is 2 × 3√3 = 6√3 m. '3√3 m' gives only one support's horizontal distance and forgets to double it for the total width. '6 m' comes from using sin 30° instead of cos 30° for the horizontal distance (6 × sin 30° = 3, doubled to 6). '12 m' comes from doubling the full sloping length of 6 m without using any trigonometry at all.
- (d) √3/2 — cos 30° is one of the exact values you must know: cos 30° = √3/2. The value 1/2 is the exact value of sin 30° (and of cos 60°), not cos 30°. The value √2/2 is the exact value of cos 45° (and of sin 45°). The value 1 is the exact value of cos 0°.
- (b) 0 — At 0°, the opposite side of the right-angled triangle has shrunk to zero length, so sin 0° = 0. 1 is the value of sin 90° (mixing up the two angles). 1/2 is the value of sin 30°. 'Undefined' is what happens for tan 90°, not sin 0° — sin 0° has a perfectly good exact value.
- (a) 0.1 m — Method: the sloping surface is the hypotenuse and the vertical rise is the side opposite the 30° angle, so rise = 4.8 × sin 30°; then compare that rise with the limit. Working: the exact value of sin 30° is one half, so the rise = 4.8 × 1/2 = 2.4 m. The limit is 2.5 m, and 2.5 − 2.4 = 0.1. Answer: the ramp is 0.1 m below the limit. Working out the rise and stopping there gives 2.4 m, which answers a question that was not asked. Dividing by sin 30° instead of multiplying gives 4.8 ÷ 0.5 = 9.6 and then 9.6 − 2.5 = 7.1 m. Treating sine as proportional to the angle, so that sin 30° is a third of sin 90°, gives 4.8 ÷ 3 = 1.6 and then 2.5 − 1.6 = 0.9 m.
- (c) √3/2 — Method: sin 60° comes from half of an equilateral triangle. Cut an equilateral triangle of side 2 down its middle: each half is right-angled, with hypotenuse 2 and base 1, and Pythagoras gives its height as √3 because 2² − 1² = 3. Working: sine is the opposite side over the hypotenuse, the side opposite the 60° angle is the height √3, and the hypotenuse is 2, so sin 60° = √3 ÷ 2. Answer: sin 60° = √3/2. Swapping sine for cosine at this angle gives cos 60° = 1/2; taking the value from the other special triangle, the right-angled isosceles one, gives sin 45° = √2/2; and reading the tangent row instead of the sine row gives tan 60° = √3.
- (c) 1/2 — Method: cos 60° comes from half of an equilateral triangle. Cut an equilateral triangle of side 2 down its middle: each half is right-angled, with hypotenuse 2, base 1 and height √3. Working: cosine is the adjacent side over the hypotenuse, the side alongside the 60° angle is the base of length 1, and the hypotenuse is 2, so cos 60° = 1 ÷ 2. Answer: cos 60° = 1/2. A candidate who assumes the cosine falls steadily from cos 0° = 1 to cos 90° = 0 puts 60° two thirds of the way along and writes 1/3; the cosine does not change at a constant rate. Reading the start of the cosine row gives cos 0° = 1, and reading its end gives cos 90° = 0.
- (c) 1/2 — sin 45° = √2/2 and cos 45° = √2/2, so sin 45° × cos 45° = √2/2 × √2/2 = 2/4 = 1/2. √2/2 comes from writing down only one of the two factors and forgetting to multiply by the other. √2 comes from adding the two exact values instead of multiplying them: √2/2 + √2/2 = √2. 1 comes from wrongly treating sin 45° × cos 45° as sin(45° + 45°) = sin 90° = 1 — multiplying two ratios is not the same as adding their angles.
- (c) 1 — sin 30° = 1/2 and cos 60° = 1/2, so sin 30° + cos 60° = 1/2 + 1/2 = 1. 0 comes from subtracting the two values instead of adding them (1/2 − 1/2). 1/2 comes from writing down only one of the two exact values and forgetting to add the other. √3 comes from swapping the two angles and working out sin 60° + cos 30° = √3/2 + √3/2 = √3.
- (c) tan 45° — Method: replace each ratio by its exact value, then compare. Working: a right-angled triangle with a 45° angle is isosceles, so its opposite and adjacent sides are equal and the tangent of 45° is exactly 1. The others are cos 30° = √3/2, about 0.87; sin 45° = √2/2, about 0.71; and cos 60° = 1/2. Answer: tan 45°, the only one of the four that reaches 1. Reading √3/2 as though it were √3, about 1.73, makes cos 30° look the largest, but the division by 2 is part of the value. Ranking by the size of the angle also fails here, because the cosine of an angle falls as the angle grows.
- (d) 1 — At 45° the right-angled isosceles triangle has both shorter sides equal, so tan 45° = opposite ÷ adjacent = 1 ÷ 1 = 1. 0 is the value of tan 0° (mixing up the two angles). √3 is the value of tan 60°. √3/3 (equivalent to 1/√3) is the value of tan 30°.
Build your own mix at the worksheet builder.