Which exact trig values do I need to memorise for GCSE Maths?

Learn sine and cosine for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$. You should also know tangent for $0^\circ$, $30^\circ$, $45^\circ$ and $60^\circ$. Check your exam board's specification because the way content is classified can differ.

What does exact value mean in trigonometry?

An exact value has not been rounded. For example, $\frac{\sqrt{2}}{2}$ is the exact value of $\sin 45^\circ$, while $0.7071$ is an approximation.

How can I remember the sine exact values?

For angles $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$, use $\frac{\sqrt{0}}{2}$, $\frac{\sqrt{1}}{2}$, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$ and $\frac{\sqrt{4}}{2}$. Simplifying gives $0$, $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$ and $1$.

Why is tan 90 degrees undefined?

Since $\tan\theta=\frac{\sin\theta}{\cos\theta}$, $\tan 90^\circ$ would require division by $\cos 90^\circ=0$. Division by zero is undefined.

Can I use a calculator for exact trig values questions?

You may be able to use a calculator on a calculator paper, but its decimal display will not usually provide the required exact form. You should know the standard values and write answers using integers, fractions or surds as appropriate.

Is one over root three an exact value for tan 30 degrees?

Yes. $\frac{1}{\sqrt{3}}$ and $\frac{\sqrt{3}}{3}$ are equivalent exact values. At GCSE, the rationalised form $\frac{\sqrt{3}}{3}$ is commonly preferred.

Exact Trig Values GCSE Maths Revision Guide

Exact trig values GCSE Maths questions test whether you know particular values of sine, cosine and tangent without relying on a calculator. You need to recognise the angles $0∘0^\circ$ , $30∘30^\circ$ , $45∘45^\circ$ , $60∘60^\circ$ and $90∘90^\circ$ , then give answers as integers, fractions or surds rather than rounded decimals.

This guide explains the complete table, shows where the values come from and demonstrates how to use them in worked examples. It also includes common mistakes and exam-style practice. You can check how the topic is classified for your specification in the GCSE Maths topics by exam board and tier.

What are exact trig values?

A trigonometric value is exact when it has not been rounded. For example,

sin⁡30∘=12\sin 30^\circ=\frac{1}{2}

is exact, while $sin⁡30∘=0.5\sin 30^\circ=0.5$ is its exact decimal equivalent. For an angle such as $45∘45^\circ$ ,

sin⁡45∘=22\sin 45^\circ=\frac{\sqrt{2}}{2}

is exact, but $0.7071$ is only an approximation.

Exact form matters because a rounded decimal loses information. If an exam question asks for an exact value, an answer containing a rounded decimal will not normally meet the instruction.

For AQA GCSE Mathematics, students are required to know the exact sine and cosine values for $0∘0^\circ$ , $30∘30^\circ$ , $45∘45^\circ$ , $60∘60^\circ$ and $90∘90^\circ$ . They must also know the tangent values for $0∘0^\circ$ , $30∘30^\circ$ , $45∘45^\circ$ and $60∘60^\circ$ . Other exam boards may organise the content differently, so use the correct specification and revision materials for your course.

The exact trig values table

This is the central table to learn.

$θ\theta$	$0∘0^\circ$	$30∘30^\circ$	$45∘45^\circ$	$60∘60^\circ$	$90∘90^\circ$
$sin⁡θ\sin\theta$	$0$	$12\frac{1}{2}$	$22\frac{\sqrt{2}}{2}$	$32\frac{\sqrt{3}}{2}$	$1$
$cos⁡θ\cos\theta$	$1$	$32\frac{\sqrt{3}}{2}$	$22\frac{\sqrt{2}}{2}$	$12\frac{1}{2}$	$0$
$tan⁡θ\tan\theta$	$0$	$33\frac{\sqrt{3}}{3}$	$1$	$3\sqrt{3}$	undefined

You may also see

tan⁡30∘=13.\tan 30^\circ=\frac{1}{\sqrt{3}}.

This is equivalent to $33\frac{\sqrt{3}}{3}$ , but rationalising the denominator gives the form most commonly expected at GCSE.

The value $tan⁡90∘\tan 90^\circ$ is undefined, not zero. This is because

tan⁡θ=sin⁡θcos⁡θ\tan\theta=\frac{\sin\theta}{\cos\theta}

and $cos⁡90∘=0\cos 90^\circ=0$ . Division by zero is undefined.

How to remember the exact trig values

The sine pattern

Write the numbers $0$ , $1$ , $2$ , $3$ and $4$ under square roots, then divide each by $2$ :

42.\sin\theta=\frac{\sqrt{0}}{2},\ \frac{\sqrt{1}}{2},\ \frac{\sqrt{2}}{2},\ \frac{\sqrt{3}}{2},\ \frac{\sqrt{4}}{2}.

After simplifying, this gives

1.0,\ \frac{1}{2},\ \frac{\sqrt{2}}{2},\ \frac{\sqrt{3}}{2},\ 1.

These correspond, in order, to $0∘0^\circ$ , $30∘30^\circ$ , $45∘45^\circ$ , $60∘60^\circ$ and $90∘90^\circ$ .

The cosine pattern

Cosine uses the same list in reverse:

0.1,\ \frac{\sqrt{3}}{2},\ \frac{\sqrt{2}}{2},\ \frac{1}{2},\ 0.

This reflects the relationship

cos⁡θ=sin⁡(90∘−θ).\cos\theta=\sin(90^\circ-\theta).

For example, $cos⁡30∘=sin⁡60∘=32\cos 30^\circ=\sin 60^\circ=\frac{\sqrt{3}}{2}$ .

Finding tangent from sine and cosine

Instead of treating tangent as a completely separate list, use

tan⁡θ=sin⁡θcos⁡θ.\tan\theta=\frac{\sin\theta}{\cos\theta}.

For example,

tan⁡60∘=sin⁡60∘cos⁡60∘=3212=3.\tan 60^\circ =\frac{\sin 60^\circ}{\cos 60^\circ} =\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} =\sqrt{3}.

It is still worth memorising the four required tangent values so that you can answer direct recall questions quickly.

Why the values are exact

The values for $30∘30^\circ$ and $60∘60^\circ$ come from an equilateral triangle. Splitting an equilateral triangle of side length $2$ in half produces a right-angled triangle with hypotenuse $2$ , shortest side $1$ and remaining side $3\sqrt{3}$ .

Using SOHCAHTOA in this triangle gives

sin⁡30∘=12,cos⁡30∘=32,tan⁡30∘=13=33.\sin 30^\circ=\frac{1}{2},\qquad \cos 30^\circ=\frac{\sqrt{3}}{2},\qquad \tan 30^\circ=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}.

The $45∘45^\circ$ values come from an isosceles right-angled triangle. If both shorter sides have length $1$ , Pythagoras gives a hypotenuse of $2\sqrt{2}$ .

Therefore,

sin⁡45∘=cos⁡45∘=12=22\sin 45^\circ=\cos 45^\circ=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}

and

tan⁡45∘=11=1.\tan 45^\circ=\frac{1}{1}=1.

Understanding these triangles provides a way to reconstruct the table if you forget a value in an exam.

Exact trig values worked examples

Worked example 1: evaluating an expression

Find the exact value of

4sin⁡30∘+2cos⁡60∘.4\sin 30^\circ+2\cos 60^\circ.

Both trig values equal $12\frac{1}{2}$ , so substitute them before simplifying:

4(12)+2(12)=2+1=3.4\left(\frac{1}{2}\right)+2\left(\frac{1}{2}\right) =2+1 =3.

The exact value is $3$ .

Worked example 2: simplifying surds

Find the exact value of

sin⁡45∘cos⁡30∘.\sin 45^\circ\cos 30^\circ.

Substitute the exact values:

22×32=64.\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2} =\frac{\sqrt{6}}{4}.

The exact answer is $64\frac{\sqrt{6}}{4}$ . Do not convert this to a decimal because the question asks for an exact value.

Worked example 3: using an exact value in a triangle

A right-angled triangle has hypotenuse $10$ cm and an angle of $60∘60^\circ$ . Find the exact length of the side opposite the $60∘60^\circ$ angle.

Using sine,

sin⁡60∘=x10.\sin 60^\circ=\frac{x}{10}.

Substitute $sin⁡60∘=32\sin 60^\circ=\frac{\sqrt{3}}{2}$ :

32=x10.\frac{\sqrt{3}}{2}=\frac{x}{10}.

Multiply by $10$ :

x=10×32=53.x=10\times\frac{\sqrt{3}}{2}=5\sqrt{3}.

The exact length is $535\sqrt{3}$ cm.

More explanations and lesson materials are available on the MathsGenie exact trig values topic page and in the exact trig values revision guide.

Featured exact trig values exam question

Without using a calculator, find the exact value of

sin⁡60∘cos⁡45∘.\frac{\sin 60^\circ}{\cos 45^\circ}.

Give your answer in its simplest form.

Substitute the exact values:

3222=32.\frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{2}}{2}} =\frac{\sqrt{3}}{\sqrt{2}}.

Rationalise the denominator:

32×22=62.\frac{\sqrt{3}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}} =\frac{\sqrt{6}}{2}.

Therefore, the exact value is

62.\boxed{\frac{\sqrt{6}}{2}}.

Try 4 more exact trig values practice questions. The MathsGenie questions include worked solutions and mark-scheme guidance.

Further exam-style practice questions

Attempt these without a calculator before checking the answers.

Find the exact value of $3cos⁡60∘3\cos 60^\circ$ .
Find the exact value of $2sin⁡45∘+tan⁡45∘2\sin 45^\circ+\tan 45^\circ$ .
Simplify $sin⁡30∘cos⁡30∘\sin 30^\circ\cos 30^\circ$ .
A right-angled triangle has an adjacent side of length $6$ cm and an angle of $30∘30^\circ$ . Find the exact hypotenuse.

Answers

$3×12=323\times\frac{1}{2}=\frac{3}{2}$ .
$2×22+1=2+12\times\frac{\sqrt{2}}{2}+1=\sqrt{2}+1$ .
$12×32=34\frac{1}{2}\times\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}$ .
$cos⁡30∘=6h\cos 30^\circ=\frac{6}{h}$ , so $32=6h\frac{\sqrt{3}}{2}=\frac{6}{h}$ . Therefore $h=123=43h=\frac{12}{\sqrt{3}}=4\sqrt{3}$ cm.

Common mistakes in exact trig questions

Giving a rounded decimal: If the question requests an exact value, keep fractions and surds in the answer.
Reversing sine and cosine: Sine increases from $0$ to $1$ across the table, while cosine decreases from $1$ to $0$ .
Writing $tan⁡90∘=0\tan 90^\circ=0$ : It is undefined because calculating it would involve division by zero.
Forgetting to simplify surds: For example, $13\frac{1}{\sqrt{3}}$ can be rationalised to $33\frac{\sqrt{3}}{3}$ .
Using the wrong angle: Read whether the question asks for $30∘30^\circ$ or $60∘60^\circ$ . Their sine and cosine values are reversed.
Using an incorrect calculator mode: GCSE trigonometry normally uses degrees, so a calculator should display degree mode when checking an answer.
Rounding during a calculation: Even when an exact answer is not required, keep exact values or full calculator accuracy until the final line.

How to revise exact trig values effectively

Start by learning the sine pattern and reversing it for cosine. Test yourself by drawing a blank table and completing it from memory, including tangent and the fact that $tan⁡90∘\tan 90^\circ$ is undefined.

Next, practise substituting the values into expressions. Include questions involving fractions, surds and right-angled triangles rather than limiting your revision to simple recall.

Finally, complete mixed questions under timed conditions. The MathsGenie GCSE Maths revision area provides exam-board resources, practice questions and past papers. If gaps elsewhere make the questions difficult, the recommended order for learning GCSE Maths topics can help you decide what to revise first.

Conclusion

The essential exact trig values come from the special angles $0∘0^\circ$ , $30∘30^\circ$ , $45∘45^\circ$ , $60∘60^\circ$ and $90∘90^\circ$ . Learn the sine sequence, reverse it for cosine and use $tan⁡θ=sin⁡θcos⁡θ\tan\theta=\frac{\sin\theta}{\cos\theta}$ to understand the tangent values.

In exam questions, substitute exact fractions and surds carefully, simplify fully and avoid rounded decimals when an exact answer is requested. Use the MathsGenie lesson, practice questions and past papers to move from recalling the table to applying it confidently in unfamiliar problems.

Sources and referenced URLs