Double Angle Formula

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.

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.

Related terms

Trigonometric Identities: Trigonometric identities are mathematical equations that hold true for all values of the variables involved, regardless of the specific angle or value substituted.

Angle Addition Formulas: The angle addition formulas express the trigonometric functions of the sum or difference of two angles in terms of the trigonometric functions of the individual angles.

Inverse Trigonometric Functions: Inverse trigonometric functions, such as arcsin, arccos, and arctan, are used to find the angle given the value of a trigonometric function.