Exact Trig Values: Edexcel GCSE Foundation maths revision notes

Exact trig values in the exam

Exact trig values are the values of sin, cos and tan for the angles 0^\circ, 30^\circ, 45^\circ, 60^\circ and 90^\circ, written exactly with fractions and square roots. For example, \sin30^\circ=\frac12 exactly, and \cos30^\circ=\frac{\sqrt3}{2}, not a rounded decimal.

They come rarely, and only as one of the last questions on Paper 1, the non-calculator paper. You may be asked to write down one exact value, such as \sin30^\circ, for 1 mark, or to use one to find a side in a right-angled triangle, for 2 marks.

Since 2022 a formulae sheet has been given in the exam, with the three trig ratios on it, but learn them anyway. The exact values are not on it, so you must learn the table on this page.

This page uses the three ratios from the Trigonometry notes. Label the sides from the angle you are using: hypotenuse (H), opposite (O) and adjacent (A).

\sin x=\frac{O}{H}\qquad\cos x=\frac{A}{H}\qquad\tan x=\frac{O}{A}

The table to learn

Here is the table to learn. It gives the exact values of sin, cos and tan for 0^\circ, 30^\circ, 45^\circ, 60^\circ and 90^\circ. \tan90^\circ has no value, so you are never asked for it.

A way to remember the sin row: write \frac{\sqrt0}{2}, \frac{\sqrt1}{2}, \frac{\sqrt2}{2}, \frac{\sqrt3}{2}, \frac{\sqrt4}{2}, then simplify, because \sqrt0=0, \sqrt1=1 and \sqrt4=2. That gives 0, \frac12, \frac{\sqrt2}{2}, \frac{\sqrt3}{2}, 1. The cos row is the same list backwards.

Notice that \sin30^\circ and \cos60^\circ are both \frac12, and \sin60^\circ and \cos30^\circ are both \frac{\sqrt3}{2}. The tan row is 0, \frac{1}{\sqrt3}, 1, \sqrt3. The one to be sure of is \tan45^\circ=1: a right-angled triangle with a 45^\circ angle has two equal shorter sides, so opposite over adjacent is 1.

Worked example 1: Reading the table

Write down the exact value of

(a) \cos45^\circ

(b) \sin60^\circ

Find the angle's column in the table, then read across from sin or cos.
(a) The cos row under 45^\circ gives \cos45^\circ=\frac{\sqrt2}{2}
Your turn. (b) Write down the exact value of \sin60^\circ

Your answer

(b) The sin row under 60^\circ gives \sin60^\circ=\frac{\sqrt3}{2}
Check with the pattern: the sin row is \frac{\sqrt0}{2}, \frac{\sqrt1}{2}, \frac{\sqrt2}{2}, \frac{\sqrt3}{2}, \frac{\sqrt4}{2}, and 60^\circ is the fourth angle, so \frac{\sqrt3}{2}
Answer: (a) \frac{\sqrt2}{2} (b) \frac{\sqrt3}{2}

Worked example 2: Using the table

Work out the value of 12\sin30^\circ-5\tan45^\circ

Replace each trig value with its exact value from the table.
Your turn. Write down the exact value of \sin30^\circ

Your answer

\sin30^\circ=\frac12
\tan45^\circ=1
12\times\frac12-5\times1=6-5
Answer: 1

Finding a side exactly

On Paper 1 you have no calculator, so a trig question there uses 30^\circ, 45^\circ or 60^\circ. Label the sides, choose sin, cos or tan as usual, then put in the exact value instead of pressing a button.

If the side you want is on the top of the fraction, multiply: \tan45^\circ=\frac{x}{11} gives x=11\times1=11. If it is on the bottom, divide: \sin30^\circ=\frac{4}{y} gives y=4\div\frac12=8. Dividing by a half is the same as doubling.

Check your answer. The hypotenuse is always the longest side. If you work out the hypotenuse and it comes out shorter than another side, you multiplied when you should have divided.

Worked example 3: A side with a whole-number answer

The diagram shows a right-angled triangle. Work out the length x. Do not use a calculator.

From the 60^\circ angle, x is the adjacent and 16 cm is the hypotenuse.
A and H, so use cos: \cos60^\circ=\frac{x}{16}, so x=16\times\cos60^\circ
\cos60^\circ=\frac12
Your turn. Work out x.

Your answer

x=16\times\frac12=8
Answer: x=8 cm

Worked example 4: A side with tan 45°

The diagram shows a right-angled triangle. Work out the length x. Do not use a calculator.

From the 45^\circ angle, x is the opposite and 13 cm is the adjacent.
O and A, so use tan: \tan45^\circ=\frac{x}{13}, so x=13\times\tan45^\circ
Your turn. Write down the value of \tan45^\circ

Your answer

\tan45^\circ=1, so x=13\times1=13
Check: the angles are 45^\circ, 45^\circ and 90^\circ, so the triangle is isosceles and its two shorter sides are equal.
Answer: x=13 cm

Worked example 5: A slide in a playground

A straight slide runs from the top of a vertical post down to level ground. The post is 3 m tall. The slide makes an angle of 30^\circ with the ground. Work out the length of the slide, L. Do not use a calculator.

The slide is the hypotenuse L. From the 30^\circ angle, the post, 3 m, is the opposite.
O and H, so use sin: \sin30^\circ=\frac{3}{L}, and \sin30^\circ=\frac12. L is on the bottom, so divide.
Your turn. Work out L.

Your answer

L=3\div\frac12=6 m
Check: the slide is the hypotenuse, so it must be longer than the 3 m post. Multiplying would give 1.5 m, which is too short.
Answer: L=6 m

Common mistakes

Not learning the table. Many students leave these questions blank; practise writing the table out until you can do it from memory.
Not simplifying. \frac{\sqrt1}{2} or \frac{\sqrt4}{4} is not accepted as an exact value; write \frac12.
Mixing up sin and cos. \sin30^\circ=\frac12 but \cos30^\circ=\frac{\sqrt3}{2}; write out the sin row pattern to check, and remember the cos row is the same list backwards.
Choosing the wrong ratio. Label O, A and H from the angle before you choose sin, cos or tan.
Multiplying when you should divide. When the unknown side is on the bottom of the fraction, divide: 4\div\frac12=8, not 2.