Exact trig values: the table, the triangles and how to use them

Exact trig values are the values of sin, cos and tan for the angles 0°, 30°, 45°, 60° and 90°, written exactly as whole numbers, fractions and surds instead of rounded decimals. For example, sin 30° = ½, cos 45° = √2/2 and tan 60° = √3. You need to know these trig values for non-calculator GCSE questions, and every one of them comes from two simple triangles.

Exact trig values table

This is the table to learn. Each value is exact: √3/2 is exactly the sine of 60°, while 0.866 is only an approximation of it.

Some textbooks and mark schemes write the same values with a surd on the bottom: √2/2 is the same number as\frac{1}{\sqrt{2}}, and √3/3 is the same as\frac{1}{\sqrt{3}}. Either form is exact. tan 90° has no value, because tan θ = sin θ ÷ cos θ and cos 90° = 0, so you would be dividing by zero.

Exact value triangles

You don't have to learn the table as a list of facts. Two triangles hold all of it, and you can sketch them in the corner of the exam paper in a few seconds. Then use SOH CAH TOA (see the SOHCAHTOA guide) to read off any value.

The 45° triangle

Draw a right-angled triangle with two sides of length 1. It is isosceles, so its other two angles are both 45°. By Pythagoras the hypotenuse is\sqrt{1^2 + 1^2} = \sqrt{2}.

From either 45° angle, the opposite and adjacent sides are both 1 and the hypotenuse is √2:

The 30° and 60° triangle

Start with an equilateral triangle with sides of 2, so every angle is 60°. Cut it in half from the top vertex. Each half is a right-angled triangle with a hypotenuse of 2, a base of 1, and angles of 30°, 60° and 90°. Pythagoras gives the height:\sqrt{2^2 - 1^2} = \sqrt{3}.

From the 30° angle at the top, the opposite side is 1 and the adjacent side is √3. From the 60° angle at the bottom, they swap:

The values for 0° and 90° come from the edges of the pattern: a triangle with a 0° angle has no height, so sin 0° = 0 and cos 0° = 1, and at 90° they swap, sin 90° = 1 and cos 90° = 0. The unit circle shows this clearly: the point at angle θ is (cos θ, sin θ).

How to remember exact trig values

If you forget the triangles, this pattern rebuilds the sin and cos rows. Write the numbers 0, 1, 2, 3, 4 under the five angles, square root each one, and divide by 2:

For tan, divide sin by cos. For tan 30°, that is ½ ÷ √3/2 = 1/√3. Check the pattern once on a calculator, so you can trust it in the exam.

Check yourself on the unit circle with this question from the Trigonometry course. It asks for exact values of sin and cos.

How to use exact trig values

Exam questions use the values in three ways: work out an expression, find a side without a calculator, or find an angle from a value. Leave answers as fractions and surds unless the question says otherwise.

Example 1: work out an expression

Find the exact value of sin 60° × cos 30°.

√3 × √3 = 3, so the surds disappear and the answer is a plain fraction.

Example 2: add values with a common denominator

Find the exact value of tan 30° + tan 60°, with a rational denominator.

Example 3: find a side

A right-angled triangle has a hypotenuse of 12 cm and an angle of 30°. Find the side opposite the 30° angle without a calculator.

Example 4: an exact length in surd form

A ramp rises from the ground at 60°. Its horizontal run is 9 m. Find the exact height of the top of the ramp.

That is about 15.6 m, but the question asked for the exact height, so the answer is 9√3 m.

Example 5: find an angle from a value

In a right-angled triangle, cos x = √3/2. Find x.

Look along the cos row of the table: √3/2 sits under 30°. So x = 30°. Working backwards like this is the non-calculator version of using cos⁻¹.

Example 6: a Higher tier mix

Show that 2 sin 45° cos 45° = sin 90°.

This is a special case of the double angle formula, sin 2θ = 2 sin θ cos θ, which you meet at A-level.

Exact values in radians (A-level)

At A-level the same angles are usually written in radians. 30° is π/6, 45° is π/4, 60° is π/3 and 90° is π/2. The values do not change:

AQA A-level Maths (section E3) expects you to know exact values of sin and cos for 0, π/6, π/4, π/3, π/2, π and multiples of these, and of tan for the same angles except π/2.

Exact values beyond 90°

Every other exact value is one of the values above with a sign. Find the reference angle (the acute angle to the x-axis), take its value from the table, then use the quadrant to choose + or −:

• sin 150° = sin 30° = ½, because sin is positive in the second quadrant.
• cos 120° = −cos 60° = −½, because cos is negative in the second quadrant.
• tan 135° = −tan 45° = −1.
• cos 210° = −cos 30° = −√3/2, because cos is negative in the third quadrant.

The unit circle guide explains the quadrants and signs in full, and you can check any angle in the Trigonometry calculator.

Common mistakes

• Swapping sin and cos for 30° and 60°. sin 30° = ½, but cos 30° = √3/2. Sketch the half equilateral triangle if you are unsure.
• Writing a decimal. 0.707 is not exact. If a question says "exact", leave the surd.
• Giving a value for tan 90°. It is undefined, not 0 and not 1.
• Getting the hypotenuse wrong. In the 45° triangle it is √2, not 2. In the 30° and 60° triangle it is 2.
• Simplifying surds wrongly. √3 × √3 = 3, but √3 + √3 = 2√3, not √6.
• Rationalising halfway. 1/√2 becomes √2/2: multiply the top and the bottom by √2.

Practice questions

1. Write down the exact value of cos 60°.
2. Find the exact value of sin 45° × tan 60°.
3. Find the exact value of 4 sin 30° + tan 45°.
4. A right-angled triangle has a hypotenuse of 20 cm and an angle of 60°. Find the exact length of the side opposite the 60° angle.
5. In a right-angled triangle, tan y = 1. Find y.

Answers: 1) ½. 2) √2/2 × √3 = √6/2. 3) 4 × ½ + 1 = 3. 4) 20 × √3/2 = 10√3 cm. 5) y = 45°.

Practise with a short timed paper

Trigonometry comes up in GCSE Maths Higher, often on the non-calculator paper, and exact values come back in radians at A-level. The daily GCSE Maths Higher and A-level Maths: Algebra & Functions papers are short timed papers of 8 questions, with a new one every day.

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