4-4-3 Double Angle Identities Explained
Key Concepts of Double Angle Identities
Double Angle Identities are trigonometric formulas that express trigonometric functions of double angles (2θ) in terms of single angles (θ). Key concepts include:
- Double Angle Identity for Sine: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
- Double Angle Identity for Cosine: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
- Double Angle Identity for Tangent: \( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \)
1. Double Angle Identity for Sine
The double angle identity for sine is derived from the sum of angles formula for sine. It states:
\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]
Example:
If \( \sin(\theta) = \frac{1}{2} \) and \( \cos(\theta) = \frac{\sqrt{3}}{2} \), then \( \sin(2\theta) = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \).
2. Double Angle Identity for Cosine
The double angle identity for cosine can be expressed in three forms, derived from the sum of angles formula for cosine. The most common form is:
\[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \]
Other forms include:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
\[ \cos(2\theta) = 1 - 2\sin^2(\theta) \]
Example:
If \( \cos(\theta) = \frac{3}{5} \) and \( \sin(\theta) = \frac{4}{5} \), then \( \cos(2\theta) = \left(\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25} \).
3. Double Angle Identity for Tangent
The double angle identity for tangent is derived from the sum of angles formula for tangent. It states:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Example:
If \( \tan(\theta) = \frac{1}{2} \), then \( \tan(2\theta) = \frac{2 \cdot \frac{1}{2}}{1 - \left(\frac{1}{2}\right)^2} = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \).
Examples and Analogies
To better understand double angle identities, consider the following analogy:
Imagine you are doubling the size of a pie. The double angle identities help you understand how the ingredients (sine, cosine, tangent) change when you double the size of the pie (angle).
Practical Applications
Understanding double angle identities is crucial for various real-world applications, such as:
- Physics for analyzing periodic motion and waves.
- Engineering for designing structures and machines.
- Computer graphics for creating 3D models and animations.