Exact Trigonometric Values
Exact Trigonometric Values
For specific angles, the trigonometric ratios have exact values expressible as simple fractions or surds. These arise naturally from equilateral and isosceles right-angled triangles and must be known precisely — calculator approximations are not acceptable when an exact value is required.
The Exact Values Table
| Angle \( \theta \) | \( \sin\theta \) | \( \cos\theta \) | \( \tan\theta \) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{\sqrt{3}} \) |
| 45° | \( \frac{1}{\sqrt{2}} \) | \( \frac{1}{\sqrt{2}} \) | 1 |
| 60° | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( \sqrt{3} \) |
| 90° | 1 | 0 | undefined |
Where They Come From
45°: Isosceles right-angled triangle with legs 1, 1. Hypotenuse \( =\sqrt{2} \). So \( \sin45°=\cos45°=\frac{1}{\sqrt{2}} \), \( \tan45°=1 \).
30° and 60°: Equilateral triangle with side 2, cut in half. Half-base = 1, full side = 2, height \( =\sqrt{3} \).
\[ \sin30°=\frac{1}{2},\quad\cos30°=\frac{\sqrt{3}}{2},\quad\sin60°=\frac{\sqrt{3}}{2},\quad\cos60°=\frac{1}{2} \]Using Exact Values in Calculations
Exact values allow answers to remain in surd form — often required in Higher questions. Substitute the exact value and simplify algebraically.
Example: find the exact length of the hypotenuse in a right triangle with adjacent side 6 cm and angle 30°.
\[ \cos30° = \frac{6}{h} \implies h = \frac{6}{\cos30°} = \frac{6}{\frac{\sqrt{3}}{2}} = \frac{12}{\sqrt{3}} = \frac{12\sqrt{3}}{3} = 4\sqrt{3} \text{ cm} \]Key Takeaways
- Learn the table — especially \( \sin30°=\frac{1}{2} \), \( \cos60°=\frac{1}{2} \), \( \tan45°=1 \), \( \tan60°=\sqrt{3} \).
- \( \sin\theta = \cos(90°-\theta) \) — sin and cos are complementary.
- \( \tan90° \) is undefined — the ratio \( \text{opp/adj} \) has a zero denominator.
- When a question asks for an exact answer, substitute the exact value and simplify (rationalise if needed).