Exact Trigonometric Values

What Are Exact Trigonometric Values?

When you use a calculator to find sin 30°, you might see 0.5. This is an exact value – a precise answer that doesn't need rounding.

However, if you find sin 40° on a calculator, you get 0.6427876097... – this number goes on forever and needs to be rounded. This is not an exact value.

For certain special angles (0°, 30°, 45°, 60°, and 90°), the trigonometric ratios have exact values that can be written as simple fractions or surds. A surd is a root that cannot be simplified to a whole number or fraction (like √2 or √3).

In exams, you must know these exact values by heart – you cannot rely on your calculator for these specific angles.

Exact Values of Sin and Cos

Here are the exact values for sine and cosine at each special angle:

Understanding the Pattern

For sine (sin x):

• At 0°, sin = 0
• At 30°, sin = 1/2
• At 45°, sin = √2/2
• At 60°, sin = √3/2
• At 90°, sin = 1

Notice that the values increase as the angle increases from 0° to 90°.

For cosine (cos x):

• At 0°, cos = 1
• At 30°, cos = √3/2
• At 45°, cos = √2/2
• At 60°, cos = 1/2
• At 90°, cos = 0

Notice that the values decrease as the angle increases from 0° to 90°. In fact, the cosine values are the sine values in reverse order!