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:
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: