What is important to know about Exact trigonometric values?

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

What is important to know about Exact trigonometric values?

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

What is important to know about Exact trigonometric values?

tan 0° = 0, tan 30° = √3/3, tan 45° = 1, tan 60° = √3, tan 90° = undefined

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Geometry & Measures · Higher

Exact trigonometric values

You must memorise the exact values of sin, cos and tan of 0°, 30°, 45°, 60° and 90° for non-calculator questions.

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Revising exact trigonometric values the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Exact trigonometric values GCSE” on the site for the complete guide.

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Key facts to remember

1sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1
2cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0
3tan 0° = 0, tan 30° = √3/3, tan 45° = 1, tan 60° = √3, tan 90° = undefined
4Remember the 30-60-90 triangle (1 : √3 : 2) and 45-45-90 triangle (1 : 1 : √2).

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Formulas

SOH CAH TOA

sin = O/H, cos = A/H, tan = O/A

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Worked examples

Example 1

Without a calculator, find the exact value of sin 60° × cos 30°.

Working

sin 60° = √3/2
cos 30° = √3/2
Product: (√3/2) × (√3/2) = 3/4

Answer3/4

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Common mistakes

✗Confusing sin 30° with sin 60°.

✗Using a calculator and giving a decimal instead of the exact surd.

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Exam tips

✓Memorise the exact-value table — it comes up on non-calculator papers every year.

✓Leave your answer as a simplified surd or fraction.

✓Draw the 30-60-90 or 45-45-90 triangle if you forget the values.

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