Exact Trigonometric Values — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the exact values of sin, cos and tan at 0°, 30°, 45°, 60° and 90°, which the specification expects you to recall without a calculator.

By the end of this guide you should be able to state any of those fifteen values, use them in right-angled triangle calculations, and give an exact answer in surd form rather than a decimal.

You should also be able to reconstruct the values from two simple triangles if recall fails, recognise which angle produced a given value, and use them in the area formula and the sine and cosine rules.

The organising idea is that these values are not memorised facts but consequences of two triangles. Cut an equilateral triangle in half and you get every value for 30° and 60°. Cut a square along its diagonal and you get every value for 45°. If the table deserts you in an exam, sketching those two triangles takes about twenty seconds and regenerates the whole thing. That is worth far more than a mnemonic, because it cannot be misremembered.

Key terms and definitions

Exact value — a value written precisely, using surds or fractions, rather than as a rounded decimal.

Surd — a root that cannot be written exactly as a fraction, such as √2 or √3.

Rationalised — written with no surd in the denominator.

Non-calculator paper — the paper on which these values are required, since a calculator would supply them otherwise.

Undefined — having no value, as with tan 90°.

Core concepts

The values

Note that tan 30° is often written as √3/3, which is the rationalised form of 1/√3 — the same number, and either is accepted. Similarly √2/2 is the rationalised form of 1/√2.

Where 30° and 60° come from

Draw an equilateral triangle with sides of 2, so every angle is 60°. Cut it in half down the middle.

The half-triangle has a hypotenuse of 2, a base of 1, and a right angle at the foot. By Pythagoras, the height is √(4 − 1) = √3.

Its angles are 90°, 60° at the base and 30° at the apex.

Now read the ratios off. For the 30° angle, the opposite side is 1 and the hypotenuse is 2, so sin 30° = 1/2. The adjacent is √3, so cos 30° = √3/2, and tan 30° = 1/√3.

For the 60° angle, opposite and adjacent swap over: sin 60° = √3/2, cos 60° = 1/2, and tan 60° = √3.

Sides of 2, 1 and √3 are all you need to remember, and even those follow from halving an equilateral triangle of side 2.

Where 45° comes from

Draw a square of side 1 and cut it along the diagonal.

The resulting triangle has two sides of 1 and, by Pythagoras, a diagonal of √2. Its angles are 90°, 45° and 45°.

So sin 45° = 1/√2 and cos 45° = 1/√2, which rationalise to √2/2. Since the opposite and adjacent are equal, tan 45° = 1.

That last value is worth understanding rather than learning: whenever opposite and adjacent are equal, their ratio must be 1, and the triangle must be isosceles with angles of 45°.

Where 0° and 90° come from

Imagine the angle in a right-angled triangle shrinking towards 0°. The opposite side shrinks to nothing while the adjacent approaches the hypotenuse.

So sin 0° = 0 and cos 0° = 1, and tan 0° = 0.

Now let the angle grow towards 90°. The opposite side approaches the hypotenuse and the adjacent shrinks to nothing.

So sin 90° = 1 and cos 90° = 0. Since tan is sin divided by cos, tan 90° would require dividing by zero, so it is undefined.

The patterns

Several patterns make the table easier to hold.

The sine values from 0° to 90° run √0/2, √1/2, √2/2, √3/2, √4/2 — which simplify to 0, 1/2, √2/2, √3/2, 1. The tidy sequence under the roots is 0, 1, 2, 3, 4.

Cos is sin reversed. Reading the cos column bottom to top gives exactly the sin column top to bottom, because cos θ = sin(90° − θ).

tan = sin ÷ cos. Every tan value can be produced by dividing the two above it, which also explains why tan 90° is undefined.

Sine increases from 0 to 1 across the range while cosine decreases from 1 to 0, which is a quick way to check a value looks plausible.

Using the values

These appear wherever a non-calculator question involves a 30°, 45° or 60° angle.

In a right-angled triangle with a hypotenuse of 12 and an angle of 30°, the opposite side is 12 × sin 30° = 12 × 1/2 = 6 exactly.

In the area formula, a triangle with sides 4 and 5 and an included angle of 30° has area ½ × 4 × 5 × 1/2 = 5 exactly.

They also appear in the sine and cosine rules, and in questions asking you to show a result — where an exact value is essential, since a decimal cannot prove an exact statement.

Working backwards from a value

Questions often give a ratio and ask for the angle, which means reading the table in reverse.

If sin θ = 1/2, then θ = 30°. If cos θ = √3/2, then θ = 30°. If tan θ = √3, then θ = 60°.

Watch the pairings, since sin and cos swap between 30° and 60°: a sine of √3/2 gives 60°, while a cosine of √3/2 gives 30°.

A value of 1/√2 or √2/2, in either sin or cos, always means 45°, because that is the only angle where the two are equal.

In the sine rule, cosine rule and area formula

These values appear well beyond right-angled triangles.

In the area formula, a triangle with sides 6 and 8 and an included angle of 30° has area ½ × 6 × 8 × 1/2 = 12 exactly.

In the cosine rule, an included angle of 60° gives cos 60° = 1/2, so the term 2bc cos A becomes simply bc — which often makes an apparently heavy calculation collapse.

An angle of 120° is also common, and cos 120° = −1/2. The negative sign means the 2bc cos A term is added rather than subtracted, which is exactly the trap noted in the cosine rule guide.

Leaving answers exact

An exact answer keeps surds and fractions rather than converting to a decimal.

So 6√3 is exact, while 10.392 is an approximation. A question saying "give your answer in exact form" or "leave your answer in surd form" is asking for the former.

Rationalise the denominator where one appears, so 1/√3 is usually written √3/3, and simplify surds fully.

Multiplying by a surd often tidies well: 10 × tan 60° = 10√3, which is exact and simpler than the decimal.

Worked examples

Example 1: Reconstructing the values

Without a calculator, find sin 60°, cos 60° and tan 60°, showing where they come from.

Sketch an equilateral triangle of side 2 and cut it in half.

The half-triangle has hypotenuse 2, base 1, and height √(2² − 1²) = √3.

The 60° angle sits at the base. Opposite it is the height √3, adjacent to it is the base 1, and the hypotenuse is 2.

So sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3/1 = √3.

Example 2: An exact side length

A right-angled triangle has a hypotenuse of 8 cm and an angle of 45°. Find the exact length of the side opposite that angle.

The opposite and hypotenuse call for sin, and the unknown is on top, so multiply.

opposite = 8 × sin 45° = 8 × √2/2.

The 8 and the 2 cancel to give 4, so the answer is 4√2 cm.

That is exact. Writing 5.66 cm would be an approximation and would lose the mark on a question asking for an exact answer.

Example 3: Identifying the angle

In a right-angled triangle, the opposite side is 5 and the adjacent side is 5. Find the angle without a calculator.

tan θ = opposite ÷ adjacent = 5 ÷ 5 = 1.

The only angle in the table with a tangent of 1 is 45°.

This makes sense: equal opposite and adjacent sides make the triangle isosceles, so the two acute angles are equal and must each be 45°.

Common mistakes and how to avoid them

Swapping sin and cos at 30° and 60°. Sketch the half-equilateral triangle; the larger angle faces the larger side, so sin 60° must be the bigger value.

Giving a decimal when an exact value is asked for. Keep surds and fractions.

Leaving a surd in the denominator. Rationalise 1/√3 to √3/3.

Thinking tan 90° is very large rather than undefined. It requires dividing by zero, so it has no value.

Forgetting these are needed on the non-calculator paper. That is the whole point of learning them.

Misreading the pattern. Sine rises from 0 to 1 and cosine falls from 1 to 0. Check any recalled value against that.

Exam technique for "Exact Trigonometric Values"

Sketch the two triangles in the margin at the start of a non-calculator paper — half an equilateral triangle of side 2, and a square of side 1 cut diagonally. Twenty seconds gives you every value for the whole paper.

Substitute the exact value into the working rather than a decimal, so that surds cancel where they can.

Rationalise denominators and simplify surds before giving a final answer.

Watch for the words exact, surd form or show that — all of them require these values rather than calculator output.

Check a recalled value against the pattern: sine increasing, cosine decreasing, tan equal to their quotient.

If asked for an angle, compare the ratio you have found with the table rather than reaching for an inverse function.

Quick revision summary

These values come from two triangles: half an equilateral triangle of side 2 gives 30° and 60°; a unit square cut along its diagonal gives 45°.

sin: 0° = 0, 30° = 1/2, 45° = √2/2, 60° = √3/2, 90° = 1.

cos: 0° = 1, 30° = √3/2, 45° = √2/2, 60° = 1/2, 90° = 0.

tan: 0° = 0, 30° = 1/√3 (or √3/3), 45° = 1, 60° = √3, 90° = undefined.

Sine values follow √0/2, √1/2, √2/2, √3/2, √4/2. Cos is sin reversed, and tan = sin ÷ cos — which is why tan 90° has no value.

Sine rises from 0 to 1; cosine falls from 1 to 0. Use that to check any value you recall.

Give answers in exact form: keep surds, rationalise denominators, and never substitute a decimal on a non-calculator paper.