GCSE MATHS • HIGHER TIER
Exact Trig Values Made Simple
Recall exact sine, cosine and tangent values for key angles and use them confidently in GCSE questions.
Clear method
Exam practice
Worked proofs
Exact Trig Values in Four Moves
1
Identify
Identify the angle and required trig ratio.
2
Recall
Recall the exact value from the standard trig table.
3
Substitute
Use the exact fraction or surd value in the calculation.
4
Simplify
Simplify the result without converting to a decimal unless asked.
What You’ll Learn
Understand what exact trig values are.
Recall the key angles 0°, 30°, 45°, 60° and 90°.
Memorise exact sine, cosine and tangent values.
Use the sine pattern and reverse it for cosine.
Evaluate expressions using exact trig values.
Recognise that tan(90°) is undefined.
Link exact values to special triangles and trig graphs.
Key Rules
Key angles
Memorise the exact trig values for 0°, 30°, 45°, 60° and 90°.
Sine & cosine
For sine use √0/2, √1/2, √2/2, √3/2, √4/2. For cosine, use the same pattern in reverse.
Tangent values
tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° is undefined.
Keep values exact
Leave answers as fractions or surds. Do not convert to decimals unless the question asks you to.
Worked Proof: Exact Trig Values
Step-by-Step Method
- 1. Identify the angle and trig ratio needed.
- 2. Recall the exact value from the trig table.
- 3. Substitute the exact fraction or surd value.
- 4. Complete the calculation and simplify carefully.
- 5. Give the final answer in exact form unless a decimal is requested.
More Worked Examples
1
Find cos(60°).
From the table:
cos(60°) = 1/2
2
Find tan(45°).
From the table:
tan(45°) = 1
3
Evaluate 4 × sin(30°).
4 × 1/2 = 2
4
Evaluate 2cos(45°).
2 × (√2/2)
= √2
= √2
- Common GCSE Mistakes
- Using decimal approximations instead of exact values.
- Mixing up sine and cosine values.
- Forgetting tan(90°) is undefined.
- Incorrectly simplifying surds.
- Not learning the trig table.
- Exam Tips
- Memorise the full exact trig table.
- Use the sine pattern and reverse it for cosine.
- Keep answers exact unless a decimal is requested.
- Remember tan(45°) = 1 and tan(90°) is undefined.
- Simplify surd expressions carefully.
- Check which trig ratio the question asks for.
Skills Used in Exact Trig Values
Trigonometry
Exact Values
Cosine
Tangent
Surds
Special Angles
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Exact Trig Values Tutorial
Step-by-step walkthroughs of Exact Trig Values with clear methods and exam tips.
Frequently Asked Questions
They are the precise sine, cosine and tangent values for special angles, written using fractions and surds instead of decimal approximations.
0°, 30°, 45°, 60° and 90°.
Use the pattern √0/2, √1/2, √2/2, √3/2, √4/2.
Use the same sine pattern in reverse order.
tan(90°) is undefined.
Exact forms avoid rounding and are required in many Higher Tier questions.