Trigonometric Table values:
In trigonometry, the values of the six trigonometric ratios at certain standard angles are used frequently to solve problems involving heights, distances, and geometric relationships. Instead of calculating these values every time, mathematicians have prepared a trigonometric table that lists the exact values of the trigonometric ratios for commonly used angles.
The standard angles are: \(0^\circ, 30^\circ, 45^\circ, 60^\circ\) and \(90^\circ\).
How are the values obtained?
The exact values of the trigonometric ratios are derived using the properties of two special right-angled triangles:
- A \(45^\circ - 45^\circ - 90^\circ\) triangle.
- A \(30^\circ - 60^\circ - 90^\circ\) triangle.
The values for \(0^\circ\) and \(90^\circ\) are obtained by observing the limiting positions of these triangles.
Trigonometric table values:
|
Trigonometric ratio
|
0° | 30° | 45° | 60° | 90° |
| \(sin \theta\) | \(0\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(cos \theta\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) | \(0\) |
| \(tan \theta\) | \(0\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) | not defined |
| \(cosec \theta\) | not defined | \(2\) | \(\sqrt{2}\) | \(\frac{2}{\sqrt{3}}\) | \(1\) |
| \(sec \theta\) | \(1\) | \(\frac{2}{\sqrt{3}}\) | \(\sqrt{2}\) | \(2\) | not defined |
| \(cot \theta\) | not defined | \(\sqrt{3}\) | \(1\) | \(\frac{1}{\sqrt{3}}\) | \(0\) |
Important!
- The values of \(sin\ \theta\) increase as the angle increases from \(0^\circ\) to \(90^\circ\).
- The values of \(cos\ \theta\) decrease as the angle increases from \(0^\circ\) to \(90^\circ\).
- \(tan\ \theta\) increases as \(\theta\) increases from \(0^\circ\) to \(90^\circ\) and is not defined at \(90^\circ\) because \(cos\ 90^\circ = 0\).
- \(cot\ \theta\) decrease as \(\theta\) increases from \(0^\circ\) to \(90^\circ\) and is not defined at \(0^\circ\) because \(sin\ 0^\circ = 0\)
- \(sec\ \theta\) and \(cosec\ \theta \) are the reciprocals of \(cos\ \theta\) and \(sin\ \theta\), respectively.