Honors Pre-Calculus

study guides for every class

that actually explain what's on your next test

Double Angle Formula

from class:

Honors Pre-Calculus

Definition

The double angle formula is a trigonometric identity that expresses the trigonometric functions of a double angle in terms of the trigonometric functions of the original angle. It is a fundamental tool used in solving trigonometric equations.

congrats on reading the definition of Double Angle Formula. now let's actually learn it.

ok, let's learn stuff

5 Must Know Facts For Your Next Test

  1. The double angle formula for sine is $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.
  2. The double angle formula for cosine is $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$.
  3. The double angle formula for tangent is $\tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}$.
  4. The double angle formulas are derived from the angle addition formulas by setting the two angles equal to each other.
  5. The double angle formulas are useful in simplifying trigonometric expressions, solving trigonometric equations, and evaluating trigonometric functions.

Review Questions

  • Explain how the double angle formulas are derived from the angle addition formulas.
    • The double angle formulas are derived from the angle addition formulas by setting the two angles equal to each other. For example, the double angle formula for sine, $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, is obtained by setting the two angles in the angle addition formula for sine, $\sin(\theta + \phi) = \sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi)$, equal to each other, i.e., $\phi = \theta$.
  • Describe the applications of the double angle formulas in solving trigonometric equations.
    • The double angle formulas are essential tools in solving trigonometric equations. They can be used to simplify complex trigonometric expressions, which can then be solved using algebraic techniques. For example, the double angle formula for cosine can be used to solve equations of the form $\cos(2\theta) = k$, where $k$ is a constant. By rearranging the formula, we can isolate $\theta$ and find the solutions to the equation.
  • Analyze how the double angle formulas can be used to evaluate trigonometric functions for special angles.
    • The double angle formulas can be used to efficiently evaluate trigonometric functions for special angles, such as $30^\circ$, $45^\circ$, and $60^\circ$. By applying the formulas recursively, we can express the trigonometric functions of these angles in terms of the functions of simpler angles, such as $0^\circ$ and $30^\circ$, which have well-known values. This allows for quick and accurate evaluation of trigonometric functions without the need for lengthy calculations.

"Double Angle Formula" also found in:

© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Glossary
Guides