Trigonometric Ratios of Specific Angles — Table & Values
Trigonometric Ratios of Specific Angles — Table & Values
TL;DR
The trigonometric ratios of specific angles are the exact values of sin, cos, tan, csc, sec, and cot at 0°, 30°, 45°, 60°, and 90° — five angles whose ratios come out as clean surds, not decimals. This article gives the full values table, shows where each number comes from (the 30-60-90 and 45-45-90 triangles plus the unit circle), explains why entries like tan 90° are undefined, and works through six examples.
What Are the Trigonometric Ratios of Specific Angles?
The trigonometric ratios of specific angles are the exact values that the six ratios — sine, cosine, tangent, cosecant, secant, and cotangent — take at the standard angles 0°, 30°, 45°, 60°, and 90°. They are called specific (or standard) angles because their ratios can be written as exact fractions and surds rather than approximate decimals, which is why they appear in nearly every textbook derivation and exam question.
Here is the master table. Memorise it eventually — but the rest of this article shows you how to rebuild it, so you never have to trust your memory alone.
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | \frac{1}{2} | \frac{1}{\sqrt{2}} | \frac{\sqrt{3}}{2} | 1 |
| cos θ | 1 | \frac{\sqrt{3}}{2} | \frac{1}{\sqrt{2}} | \frac{1}{2} | 0 |
| tan θ | 0 | \frac{1}{\sqrt{3}} | 1 | \sqrt{3} | undefined |
| csc θ | undefined | 2 | \sqrt{2} | \frac{2}{\sqrt{3}} | 1 |
| sec θ | 1 | \frac{2}{\sqrt{3}} | \sqrt{2} | 2 | undefined |
| cot θ | undefined | \sqrt{3} | 1 | \frac{1}{\sqrt{3}} | 0 |
Where do these exact values come from?
Two triangles and one circle generate the whole table. The 45° column comes from a right isosceles triangle; the 30° and 60° columns come from half an equilateral triangle; the 0° and 90° edge values come from the unit circle, where they are read off as coordinates.
Examples of Trigonometric Ratios of Specific Angles
Example 1
Evaluate sin 30° + cos 60°.
Read both values from the table:
sin 30° = \frac{1}{2}, cos 60° = \frac{1}{2}
sin 30° + cos 60° = \frac{1}{2} + \frac{1}{2} = 1.
Example 2
Evaluate tan 45° − tan 30° without a calculator.
tan 45° − tan 30° = 1 − \frac{1}{\sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}}.
Example 3
Show that sin² 60° + cos² 60° = 1.
Substitute the table values for 60°: sin 60° = \frac{\sqrt{3}}{2}, cos 60° = \frac{1}{2}.
sin² 60° + cos² 60° = (\frac{\sqrt{3}}{2})² + (\frac{1}{2})² = \frac{3}{4} + \frac{1}{4} = 1.
Example 4
Evaluate \frac{\tan 60° - \tan 30°}{1 + \tan 60° \cdot \tan 30°}. Substitute tan 60° = \sqrt{3} and tan 30° = \frac{1}{\sqrt{3}}:
The result is \frac{\sqrt{3}}{2}.
Key Takeaways
- The trigonometric ratios of specific angles are the exact sin, cos, tan, csc, sec, and cot values at 0°, 30°, 45°, 60°, and 90°.
- The 45° values come from a 45-45-90 triangle; the 30° and 60° values from half an equilateral triangle; the 0° and 90° values are read as coordinates on the unit circle.