Half-angle formulas

Definition

Half-angle formulas are trigonometric identities that express the sine, cosine, and tangent of half an angle in terms of the square root. These formulas are useful for simplifying expressions involving trigonometric functions.

5 Must Know Facts For Your Next Test

1. The half-angle formula for sine is $\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}$.
2. The half-angle formula for cosine is $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$.
3. The half-angle formula for tangent is $\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$ or alternatively $= \frac{1 - \cos(\theta)}{\sin(\theta)}$.
4. The sign of the result from the half-angle formulas depends on the quadrant in which $\frac{θ}{2}$ lies.
5. These formulas are derived from the double-angle formulas and can be used to solve trigonometric equations.

Review Questions

• What is the half-angle formula for cosine?
• How do you determine the sign of a result when using a half-angle formula?
• Which identities can be used to derive the half-angle formulas?

Related terms

Double-Angle Formulas: Trigonometric identities that express trigonometric functions of double angles ($2θ$) in terms of single angles ($θ$).

Reduction Formulas: Formulas that simplify trigonometric expressions by reducing angles greater than $90^°$ to equivalent expressions with angles less than or equal to $90^°$.

\textitPythagorean Identities: \textitTrigonometric identities based on Pythagoras' theorem, such as $sin^2(x) + cos^2(x) = 1$.