Half Angle Identity

Definition

The half angle identity is a trigonometric identity that expresses the sine and cosine of half an angle in terms of the sine and cosine of the original angle. This identity is useful for simplifying expressions and solving equations that involve angles that are multiples of known angles, making it easier to handle calculations that involve multiple angles.

5 Must Know Facts For Your Next Test

1. The half angle identities for sine and cosine can be expressed as: $$ ext{sin}\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \text{cos}(\theta)}{2}}$$ and $$\text{cos}\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \text{cos}(\theta)}{2}}$$.
2. The sign (positive or negative) of the half angle identities depends on the quadrant in which the half-angle resides.
3. Half angle identities are particularly useful when trying to solve equations involving trigonometric functions with angles that are not standard angles like 30°, 45°, or 60°.
4. These identities can help in integration problems in calculus where angles are halved in definite integrals.
5. Using half angle identities can simplify expressions significantly, allowing for easier manipulation and solving of trigonometric equations.

Review Questions

• How do you derive the half angle identities from the double angle identities?
  • To derive the half angle identities from the double angle identities, start with the double angle formulas: $$\text{sin}(2\theta) = 2\text{sin}(\theta)\text{cos}(\theta)$$ and $$\text{cos}(2\theta) = \text{cos}^2(\theta) - \text{sin}^2(\theta)$$. By letting $$\theta$$ represent half of an angle, such as $$x$$ (where $$x = 2\theta$$), you can rearrange these formulas to express $$\text{sin}(\frac{x}{2})$$ and $$\text{cos}(\frac{x}{2})$$ in terms of $$x$$, leading to the established half angle identities.
• What role do the half angle identities play in solving trigonometric equations involving multiple angles?
  • Half angle identities allow us to break down complex trigonometric equations involving multiple angles into simpler components. By converting larger angles into half-angle terms, it becomes easier to manipulate and solve equations. This is especially beneficial when dealing with angles that result from compositions or combinations of known angles, helping to create solvable expressions without requiring numerical approximation.
• Evaluate the significance of using half angle identities in calculus, particularly in integration problems.
  • In calculus, using half angle identities is significant because they simplify integrals that involve trigonometric functions by converting them into more manageable forms. For example, when integrating expressions like $$\int \text{sin}^2(x) dx$$ or $$\int \text{cos}^2(x) dx$$, applying half angle identities allows for easier integration through substitution. This simplification can lead to straightforward solutions and reduce computation time, demonstrating the practical utility of these identities in advanced mathematical applications.

Related terms

Double Angle Identity: A trigonometric identity that expresses the sine and cosine of double an angle in terms of the sine and cosine of the original angle.

Pythagorean Identity:

An essential trigonometric identity derived from the Pythagorean theorem, stating that for any angle $$ heta$$, $$ ext{sin}^2( heta) + ext{cos}^2( heta) = 1$$.

Trigonometric Functions:

Functions such as sine, cosine, and tangent that relate the angles of a triangle to the lengths of its sides.