Half-Angle Formulas

Definition

Half-angle formulas are a set of trigonometric identities that allow you to express the sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the full angle. These formulas are particularly useful in solving trigonometric equations and simplifying complex trigonometric expressions.

5 Must Know Facts For Your Next Test

1. Half-angle formulas are derived from the double-angle formulas by solving for the trigonometric functions of half the angle.
2. The half-angle formulas are particularly useful in solving trigonometric equations and simplifying complex trigonometric expressions.
3. Half-angle formulas can be used to express the sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the full angle.
4. Understanding and applying half-angle formulas is essential in solving problems related to 7.3 Double-Angle, Half-Angle, and Reduction Formulas and 7.5 Solving Trigonometric Equations.
5. Half-angle formulas can be used to find the values of trigonometric functions for angles that are not easily measured or calculated, such as 15°, 30°, or 75°.

Review Questions

• Explain how half-angle formulas are derived from double-angle formulas and how they are used to simplify trigonometric expressions.
  • Half-angle formulas are derived from double-angle formulas by solving for the trigonometric functions of half the angle. For example, the double-angle formula for cosine, $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$, can be rearranged to solve for $\cos(\theta)$, resulting in the half-angle formula $\cos(\theta) = \pm\sqrt{\frac{1 + \cos(2\theta)}{2}}$. These half-angle formulas are then used to simplify complex trigonometric expressions by expressing the functions of half an angle in terms of the functions of the full angle, which can be more easily calculated or measured.
• Describe how half-angle formulas are used to solve trigonometric equations, and provide an example of how they can be applied.
  • Half-angle formulas are essential in solving trigonometric equations, as they allow you to rewrite the equations in terms of the full angle, which can then be solved more easily. For example, consider the equation $\sin(2\theta) = \frac{1}{2}$. Using the half-angle formula $\sin(\theta) = \frac{2\sin(\theta/2)\cos(\theta/2)}{1}$, we can rewrite the equation as $2\sin(\theta/2)\cos(\theta/2) = \frac{1}{2}$. Solving this equation, we can find the values of $\theta/2$ that satisfy the original equation, and then use the half-angle formula to find the values of the full angle $\theta$.
• Analyze how the understanding of half-angle formulas can be applied to find the values of trigonometric functions for angles that are not easily measured or calculated, and explain the significance of this application.
  • The understanding of half-angle formulas allows you to find the values of trigonometric functions for angles that are not easily measured or calculated, such as 15°, 30°, or 75°. This is because half-angle formulas express the trigonometric functions of half an angle in terms of the functions of the full angle, which can be more easily determined. For example, the half-angle formula for cosine, $\cos(\theta) = \pm\sqrt{\frac{1 + \cos(2\theta)}{2}}$, can be used to find the value of $\cos(15°)$ by first calculating $\cos(30°)$ and then applying the half-angle formula. This ability to find the values of trigonometric functions for non-standard angles is crucial in many applications, such as engineering, physics, and mathematics, where precise trigonometric calculations are required.

Related terms

Double-Angle Formulas:

Double-angle formulas are trigonometric identities that express the sine, cosine, and tangent of twice an angle in terms of the trigonometric functions of the original angle.

Reduction Formulas:

Reduction formulas are trigonometric identities that allow you to express trigonometric functions of an angle in terms of the functions of a smaller angle.

Trigonometric Equations: Trigonometric equations are equations that involve trigonometric functions, such as sine, cosine, and tangent, and need to be solved to find the values of the variables.