Exact trigonometric values
Derive special-angle ratios and use exact fractions and surds without a calculator.
Section Paper 2 · Extended · Non-calculator
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Simplify surds and fractions.
- Use right-triangle ratios.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is √2 divided by 2?
What you will learn
- Know sine and cosine at 0°, 30°, 45°, 60° and 90°.
- Know tangent at 0°, 30°, 45° and 60°.
Two special triangles explain the values
Cutting an equilateral triangle of side two in half gives a 30–60–90 triangle with sides 1, √3 and 2. Therefore sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2 and cos 60° = 1/2. Its tangent ratios are tan 30° = 1/√3 = √3/3 and tan 60° = √3.
A right isosceles triangle with legs one has hypotenuse √2. Thus sin 45° = cos 45° = 1/√2 = √2/2 and tan 45° = 1. These are exact values; replacing √2/2 by 0.707 loses exactness. A sketch of each triangle can recover the values if memory fails.
Include the endpoint angles
On the unit circle or from limiting right-triangle positions, sin 0° = 0, cos 0° = 1, sin 90° = 1 and cos 90° = 0. Across 0°, 30°, 45°, 60°, 90°, the sine values are 0, 1/2, √2/2, √3/2, 1; the cosine values reverse this order.
Tangent is sine divided by cosine: tan 0° = 0, tan 30° = √3/3, tan 45° = 1 and tan 60° = √3. Tangent 90° is undefined because its cosine denominator is zero; it is not one of the required exact tangent values. Use the exact ratios to solve simple non-calculator triangle problems.
Pause and explain
What is exact cos 60°?
Worked example
A right triangle has hypotenuse 12 cm and angle 60°. Find the opposite side exactly.
Show the worked solution
- The ratio is sin 60° = opposite/12.
- Use exact sin 60° = √3/2.
- Opposite = 12 × √3/2 = 6√3 cm.
Answer 6√3 cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find sin 45° + cos 45° exactly.
Give me a hint
The two ratios are equal.
Compare my reasoning
- Each ratio is √2/2.
- Their sum is 2 × √2/2.
- Simplify to √2 without a decimal.
√2.
Look for these in your work
- Used exact special-angle values.
- Combined equal surd terms.
Find tan 30° × tan 60° exactly.
Give me a hint
Use √3/3 and √3.
Compare my reasoning
- The product is (√3/3) × √3.
- The numerator becomes three.
- Three divided by three is one.
1.
Look for these in your work
- Used the two exact tangent values.
- Simplified the surd product.
A ramp has hypotenuse 10 m and slope 45°. Find its horizontal and vertical components exactly.
Give me a hint
Sine and cosine forty-five are equal.
Compare my reasoning
- Each component is 10 × √2/2.
- Both simplify to 5√2 m.
- Their squared sum is 50 + 50 = 100, verifying the hypotenuse.
Both components are 5√2 m.
Look for these in your work
- Retained exactness.
- Verified the two components with Pythagoras.
Common mistakes
- Interchanging sine sixty and cosine sixty.
- Replacing an exact requested answer with a rounded decimal.
- Assigning a finite value to tangent ninety.
What can you explain now?
Is tan 90° a finite value?
Compare with the explanation
No; it is undefined.
Tangent divides sine by cosine, and cos 90° = 0 would require division by zero.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, reconstruct the sine and cosine rows from the two special triangles, including endpoint values.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.