1. What Are Multiple Angle Identities?
Multiple angle identities tell us how to express trigonometric ratios of angles like \(2\theta\), \(3\theta\), or \(n\theta\) in terms of ratios of \(\theta\). These identities simplify expressions and help evaluate non-standard angles without calculators.
2. Double Angle Identities for sin 2θ, cos 2θ and tan 2θ
These three formulas are the most important multiple-angle identities.
\( \sin 2\theta = 2 \sin \theta \cos \theta \)
\( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
\( = 1 - 2\sin^2 \theta \)
\( = 2\cos^2 \theta - 1 \)
\( \tan 2\theta = \dfrac{2 \tan \theta}{1 - \tan^2 \theta} \)
2.1. Examples
Example 1: If \( \sin \theta = \dfrac{3}{5} \) and \( \cos \theta = \dfrac{4}{5} \), then:
\( \sin 2\theta = 2 \cdot \dfrac{3}{5} \cdot \dfrac{4}{5} = \dfrac{24}{25} \)
Example 2: If \( \cos \theta = \dfrac{4}{5} \):
\( \cos 2\theta = 2\cos^2 \theta - 1 = 2 \cdot \dfrac{16}{25} - 1 = \dfrac{7}{25} \)
3. Triple Angle Identities
Triple-angle identities express \(\sin 3\theta\), \(\cos 3\theta\), and \(\tan 3\theta\) in terms of \(\theta\). These are especially useful in algebraic manipulation.
\( \sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta \)
\( \cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta \)
\( \tan 3\theta = \dfrac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta} \)
3.1. Example
If \( \sin \theta = \dfrac{1}{2} \), then:
\( \sin 3\theta = 3 \cdot \dfrac{1}{2} - 4 \cdot \dfrac{1}{8} = \dfrac{3}{2} - \dfrac{1}{2} = 1 \)
4. Other Useful Multiple Angle Forms
Sometimes expressions need alternate forms. These include:
\( \sin^2 \theta = \dfrac{1 - \cos 2\theta}{2} \)
\( \cos^2 \theta = \dfrac{1 + \cos 2\theta}{2} \)
\( \sin^2 \theta - \cos^2 \theta = -\cos 2\theta + \cos 2\theta = \cdots \)
These often appear when simplifying integrals or reducing powers of sin and cos.
5. Using Multiple Angle Identities to Evaluate Angles
Identities like sin 2θ or sin 3θ help evaluate angles such as 2×30°, 3×30°, 3×45° directly.
5.1. Example
Evaluate \( \sin 90^\circ \) using the identity \( \sin 3\theta \).
Let \( 3\theta = 90^\circ \Rightarrow \theta = 30^\circ \)
\( \sin 3\theta = 1 = 3\sin 30^\circ - 4\sin^3 30^\circ \)
\( = 3 \cdot \dfrac{1}{2} - 4 \cdot \dfrac{1}{8} = 1 \)