1. Why Special Angles Are Important
The angles 0°, 30°, 45°, 60°, and 90° appear repeatedly in trigonometry. Their values are standard and do not depend on any specific triangle.
These values are used in simplifying expressions, solving equations, proving identities, and solving geometry/trigonometry problems quickly.
2. Special Angle Values for sin and cos
The easiest way to memorise sin and cos values is through patterns:
- sin values follow the pattern: \(0, \dfrac{1}{2}, \dfrac{1}{\sqrt{2}}, \dfrac{\sqrt{3}}{2}, 1\)
- cos values are just the reverse
Here is the complete table:
| Angle | sin \(\theta\) | cos \(\theta\) |
|---|---|---|
| 0° | 0 | 1 |
| 30° | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) |
| 45° | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{1}{\sqrt{2}}\) |
| 60° | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) |
| 90° | 1 | 0 |
2.1. Quick Pattern for sin Values
You can memorise sin values by using:
\( \sin \theta = \sqrt{0/4},\; \sqrt{1/4},\; \sqrt{2/4},\; \sqrt{3/4},\; \sqrt{4/4} \)
for \(0°, 30°, 45°, 60°, 90°\).
This gives: 0, 1/2, 1/√2, √3/2, 1.
3. Values of tan and cot at Special Angles
tan values can be obtained using:
\( \tan \theta = \dfrac{\sin \theta}{\cos \theta} \)
cot values are the reciprocals of tan.
| Angle | tan \(\theta\) | cot \(\theta\) |
|---|---|---|
| 0° | 0 | Undefined |
| 30° | \(\dfrac{1}{\sqrt{3}}\) | \(\sqrt{3}\) |
| 45° | 1 | 1 |
| 60° | \(\sqrt{3}\) | \(\dfrac{1}{\sqrt{3}}\) |
| 90° | Undefined | 0 |
4. Values of sec and cosec
sec and cosec are reciprocals of cos and sin respectively.
| Angle | sec \(\theta\) | cosec \(\theta\) |
|---|---|---|
| 0° | 1 | Undefined |
| 30° | \(\dfrac{2}{\sqrt{3}}\) | 2 |
| 45° | \(\sqrt{2}\) | \(\sqrt{2}\) |
| 60° | 2 | \(\dfrac{2}{\sqrt{3}}\) |
| 90° | Undefined | 1 |
5. Geometric Derivation for 30°–60° Values
Using an equilateral triangle of side \(2a\), altitude \(a\sqrt{3}\), we can derive 30° and 60° values:
- \(\sin 30° = 1/2\)
- \(\cos 30° = \sqrt{3}/2\)
- \(\sin 60° = \sqrt{3}/2\)
- \(\cos 60° = 1/2\)
6. Special Angle Table (All Ratios Together)
Full reference table for all six ratios:
| Angle | sin | cos | tan | cot | sec | cosec |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Undefined | 1 | Undefined |
| 30° | 1/2 | √3/2 | 1/√3 | √3 | 2/√3 | 2 |
| 45° | 1/√2 | 1/√2 | 1 | 1 | √2 | √2 |
| 60° | √3/2 | 1/2 | √3 | 1/√3 | 2 | 2/√3 |
| 90° | 1 | 0 | Undefined | 0 | Undefined | 1 |