Double Angle Trigonometric Identities

What are Double Angle Identities?

Double angle identities are trigonometric formulas that express the sine, cosine, and tangent of twice an angle (2θ) in terms of trigonometric functions of the original angle (θ). These identities are crucial in various mathematical and physical applications, simplifying complex trigonometric expressions and solving advanced problems.

The Double Angle Formulas

The primary double angle identities are:

\[ \begin{align*} \sin 2\theta &= 2\sin \theta \cos \theta \\ \cos 2\theta &= \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta \\ \tan 2\theta &= \frac{2\tan \theta}{1 - \tan^2 \theta} \end{align*} \]

Where:

• \(\theta\) is the original angle
• \(2\theta\) is twice the original angle
• \(\sin\), \(\cos\), and \(\tan\) are the sine, cosine, and tangent functions, respectively

Derivation and Explanation

These identities can be derived using the sum formulas for sine and cosine, considering that 2θ = θ + θ. The tangent identity is derived from the quotient of sine and cosine double angle formulas.

Example Calculation

Let's calculate the double angle identities for θ = 30°:

\[ \begin{align*} \sin 30° &= \frac{1}{2}, \cos 30° = \frac{\sqrt{3}}{2}, \tan 30° = \frac{1}{\sqrt{3}} \\[10pt] \sin 60° &= 2\sin 30° \cos 30° = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \\[10pt] \cos 60° &= \cos^2 30° - \sin^2 30° = \left(\frac{\sqrt{3}}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{1}{2} \\[10pt] \tan 60° &= \frac{2\tan 30°}{1 - \tan^2 30°} = \frac{2 \cdot \frac{1}{\sqrt{3}}}{1 - \left(\frac{1}{\sqrt{3}}\right)^2} = \sqrt{3} \end{align*} \]

Visual Representation

This diagram illustrates the relationship between θ (30°) and 2θ (60°) on the unit circle, visually representing the double angle concept.