key term - Half-Angle Formulas

Definition

Half-angle formulas are trigonometric identities that express the sine, cosine, and tangent of half the angle in terms of the sine, cosine, and tangent of the full angle. These formulas are essential for verifying trigonometric identities, simplifying trigonometric expressions, and solving trigonometric equations.

5 Must Know Facts For Your Next Test

1. The half-angle formulas for sine, cosine, and tangent are derived from the angle addition formulas.
2. Half-angle formulas are particularly useful for simplifying trigonometric expressions involving angles that are half the size of a reference angle.
3. These formulas can be used to verify trigonometric identities by expressing both sides of the identity in terms of the same trigonometric functions.
4. Half-angle formulas play a crucial role in solving trigonometric equations, as they allow you to transform the equation into a simpler form that can be solved more easily.
5. Understanding and applying half-angle formulas is essential for success in topics such as trigonometric function analysis, trigonometric equation solving, and trigonometric identity verification.

Review Questions

• Explain how the half-angle formulas are derived from the angle addition formulas.
  • The half-angle formulas are derived from the angle addition formulas by setting the second angle to half the value of the first angle. For example, the formula for $\sin(\frac{x}{2})$ can be obtained by setting $y = \frac{x}{2}$ in the angle addition formula for sine: $\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)$. Simplifying this expression and substituting $y = \frac{x}{2}$ gives the half-angle formula for sine.
• Describe how half-angle formulas can be used to verify trigonometric identities.
  • To use half-angle formulas to verify a trigonometric identity, you would express both sides of the identity in terms of the same trigonometric functions using the half-angle formulas. For example, to verify the identity $\sin^2(x) + \cos^2(x) = 1$, you could use the half-angle formulas to rewrite both sides in terms of $\sin(x)$ and $\cos(x)$. If the two sides are equivalent after this transformation, the identity is verified.
• Analyze how half-angle formulas can be applied to solve trigonometric equations.
  • Half-angle formulas are particularly useful for solving trigonometric equations because they allow you to transform the equation into a simpler form that can be solved more easily. For instance, if a trigonometric equation involves an angle that is half the size of a reference angle, you can use the half-angle formulas to rewrite the equation in terms of the reference angle. This can then be solved using techniques such as factoring, quadratic formula, or inverse trigonometric functions. The ability to manipulate the equation using half-angle formulas is a crucial step in finding the solutions to many trigonometric equations.

Related terms

Trigonometric Identities:

Trigonometric identities are mathematical equations that hold true for all values of the variables involved, often used to simplify and manipulate trigonometric expressions.

Angle Addition Formulas:

Angle addition formulas express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine, cosine, and tangent of the individual angles.

Inverse Trigonometric Functions:

Inverse trigonometric functions, such as arcsin, arccos, and arctan, allow you to find the angle given the value of a trigonometric function.