Exact values mean answers left with surds and fractions, not rounded decimals. The skill appears whenever a question says “exact”, “without a calculator” or “give your answer in surd form”, and it is the base for the rest of advanced non-calculator reasoning.
Where do the exact values come from?
Draw a square of side 1 and cut it along a diagonal.
You get a right isosceles triangle with sides 1, 1 and √2. Draw an equilateral triangle of side 2 and cut it in half. You get a right triangle with sides 1, √3 and 2, with angles 30° and 60°.
Reading ratios from these sketches gives the values:
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Note that sin 30° = cos 60° and sin 60° = cos 30°. Complementary angles swap sine and cosine.
How do you rationalise a denominator?
If the denominator is a single surd such as √5, multiply top and bottom by √5. So 10/√5 = 10√5/5 = 2√5.
If the denominator is a sum such as √3 − 1, multiply top and bottom by the conjugate √3 + 1. Then (√3 − 1)(√3 + 1) = 3 − 1 = 2, a whole number.
Worked example
Without a calculator, find the exact value of (tan 60° + 1)/(tan 60° − 1), giving your answer in the form a + b√3.
Step 1, substitute the exact value: tan 60° = √3, so the expression is (√3 + 1)/(√3 − 1).
Step 2, rationalise: multiply top and bottom by √3 + 1.
Step 3, bottom: (√3 − 1)(√3 + 1) = 3 − 1 = 2.
Step 4, top: (√3 + 1)² = 3 + 2√3 + 1 = 4 + 2√3.
Step 5, divide: (4 + 2√3)/2 = 2 + √3.
Check: with √3 ≈ 1.732, the original is 2.732/0.732 ≈ 3.732, and 2 + √3 ≈ 3.732. They agree.
The mistake to watch for
A common slip is to square a bracket without the middle term.
Mistaken working: (√3 + 1)² = 3 + 1 = 4
The student squared each term and forgot 2 × √3 × 1.
The correction is to expand (√3 + 1)(√3 + 1) in full, giving 3 + √3 + √3 + 1 = 4 + 2√3. A quick decimal check exposes the error: 2.732² is about 7.46, not 4.
Check yourself
Try these without a calculator, then open each answer.
1. Find the exact value of cos 30° × tan 60°.
Show answer
cos 30° = √3/2 and tan 60° = √3. The product is (√3 × √3)/2 = 3/2.
3/2
2. Rationalise 10/√5.
Show answer
Multiply top and bottom by √5: 10√5/5 = 2√5. Check: 10/2.236 ≈ 4.472 and 2√5 ≈ 4.472.
3. Find the exact value of sin 45° × cos 45°.
Show answer
Each is √2/2, so the product is (√2/2)² = 2/4 = 1/2.
Where this leads next
With exact values secure, go on to choosing algebra before numerical substitution. The triangle reasoning board and the non-calculator working trainer let you test your exact answers.
Some students can quote the table but freeze when it appears inside a longer question. Our teachers look for exactly where that happens in online one-to-one Additional Mathematics tuition.