Exact trig values – GCSE Maths

Foundation and Higher · Grade 5 · Non-calculator

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You are expected to know sin, cos and tan of $0^{\circ },30^{\circ },45^{\circ },60^{\circ },90^{\circ }$ exactly, with no calculator. Two triangles rebuild them: half an equilateral (sides $1,\sqrt{3},2$) and an isosceles right-angled triangle (sides $1,1,\sqrt{2}$). $\sin 30^{\circ }=\frac{1}{2}$, $\cos 30^{\circ }=\frac{\sqrt{3}}{2}$, $\tan 45^{\circ }=1$, $\sin 90^{\circ }=1$, $\cos 90^{\circ }=0$.

A non-calculator side in a right-angled triangle with $30^{\circ }$ or $60^{\circ }$ is this table plus SOHCAHTOA, leaving surds in the answer. $\tan 90^{\circ }$ is undefined — do not write a number for it.

Learn $\sin /\cos /an$ of $0,30,45,60,90$. If you forget, draw the $45$-$45$-$90$ (sides $1,1,\sqrt{2}$) or $30$-$60$-$90$ (sides $1,\sqrt{3},2$) triangle. Non-calc side lengths use these, not a calculator.

Objectives

• Recall the exact value of sine at 30, 45 and 60.
• Recall the exact value of cosine and of tangent.
• Use sin θ = cos(90 − θ) to name the matching angle.

Key points

• Recall the exact values of $\sin$, $\cos$ and $\tan$ for $0^{\circ }$, $30^{\circ }$, $45^{\circ }$, $60^{\circ }$ and $90^{\circ }$.
• Recognise an exact value and state the angle, e.g. $\cos \theta =\frac{\sqrt{3}}{2}$ gives $\theta =30^{\circ }$.
• Use exact values to find sides in right-angled triangles without a calculator, leaving surds in the answer.
• Rebuild the values quickly from the two special triangles if you forget them.

Worked examples

Questions and answers

GCSE Maths exact trig values questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Writing a rounded decimal like 0.866 instead of √3/2

   Exact values require exact surds or fractions, never decimals

2. Swapping sin 30 and cos 30

   sin 30 = 1/2 and cos 30 = √3/2

3. Leaving tan 30 as 1/√3 when rationalised form is requested

   Multiply top and bottom by √3 to get √3/3

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Sketch the two special triangles in the margin at the start of the non-calculator paper — it takes 30 seconds and removes the memory risk.
• If a question is on the non-calculator paper and involves $30^{\circ }$, $45^{\circ }$ or $60^{\circ }$, the examiner expects an exact surd answer.

Printable worksheet

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Questions students ask

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More on Exact trig values

• Exact trig values worked examples and method
  The method in order, key facts and the full lesson explanation
• Exact trig values practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.