Exact values of sin and cos for special angles
GCSE Higher students must memorise the exact values of sin, cos and tan for the angles 0°, 30°, 45°, 60°, 90° — usually given without calculator.
The exact values table
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 (or √3/3) | 1 | √3 | undefined |
Patterns to remember
- sin and cos are mirror images: sin θ = cos (90° − θ).
- 30°, 45°, 60° all involve √3 or √2 in some form.
- 0° and 90° give the cleanest values (0 or 1).
Memory aid: hand trick
For sin: hold 5 fingers and assign them to 0, 30, 45, 60, 90.
- sin θ = √(finger number) / 2.
- 0/2 = 0, √1/2 = 1/2, √2/2, √3/2, √4/2 = 1.
For cos: same but read backwards.
✦Worked example— Worked examples
Example 1. Find sin 60° + cos 30° exactly.
- sin 60° = √3/2; cos 30° = √3/2.
- Sum = √3.
Example 2. A right-angled triangle has hypotenuse 8 and angle 30°. Find opposite (exactly).
- O = 8 × sin 30° = 8 × 1/2 = 4.
Example 3. Find tan 45° × cos 60°.
- = 1 × 1/2 = 1/2.
Where they appear
- Non-calculator papers, especially Higher tier.
- Exact-form answers in trigonometry.
- Sine/cosine rule problems where one angle is special.
⚠Common mistakes
- Confusing sin 60° with sin 30° — write the table somewhere safe.
- Forgetting tan 90° is undefined (vertical line, gradient infinite).
- Mis-rationalising 1/√3 — accept either 1/√3 or √3/3.
- Substituting decimal approximations when exact form is wanted.
- Treating sin 45° as 0.5 — it's √2/2 ≈ 0.707, NOT 1/2.
➜Try this— Quick check
Find sin² 30° + cos² 30° exactly.
- (1/2)² + (√3/2)² = 1/4 + 3/4 = 1 (Pythagorean identity).
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