Double-Angle Formulas

Let α be an angle. The double-angle (2α) trigonometric ratios can be expressed as a function of the trigonometric ratios of the angle α, that is, the double-angle formulas:

• Double-Angle Formula for Sine:
  
• Double-Angle Formula for Cosine:
  
• Double-Angle Formula for Tangent:
  

The double-angle identities are easily derived from the sum identities. Just substitute β for α.

Example

Let α be an angle equal to 30º. The trigonometric ratios of its double angle are:

• Sine of a Double-Angle (2⋅30º):
  
• Cosine of a Double-Angle (2⋅30º):
  
• Tangent of a Double-Angle (2⋅30º):
  

These results correspond to the trigonometric ratios of a 60º angle.

How Do We Get the Double-Angle Formula?

Sine of a Double Angle

Recall the angle sum formula for the sine:

If we replace β with α, we obtain the sine of a double-angle:

Cosine of a Double Angle

From the cosine sum formula we can obtain the double-angle formula:

If we substitute β for α we get the formula:

Tangent of a Double Angle

Recall the tangent sum formula:

If we substitute β for α we get the tangent double-angle formula: