The double angle formulae allow us to express trigonometric functions of \( 2\theta \) in terms of functions of \( \theta \). These are useful in simplifying trigonometric expressions, solving equations, and calculus applications.
Starting from \(\sin(A+B) = \sin A \cos B + \cos A \sin B\), if we let \( A = B = \theta \), then \[ \sin(2\theta) = \sin(\theta + \theta) = \sin\theta\cos\theta + \cos\theta\sin\theta = 2\sin\theta\cos\theta. \]
â Formula: \(\displaystyle \sin(2\theta) = 2\sin\theta\cos\theta\).
Using \(\cos(A+B) = \cos A \cos B - \sin A \sin B\), \[ \cos(2\theta) = \cos^2\theta - \sin^2\theta. \] We can also rewrite using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\): \[ \cos(2\theta) = 1 - 2\sin^2\theta \] or \[ \cos(2\theta) = 2\cos^2\theta - 1. \]
â Formulas: \[ \cos(2\theta) = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1. \]
Using \(\tan(2\theta) = \dfrac{\sin(2\theta)}{\cos(2\theta)}\), and substituting the double angle expressions: \[ \tan(2\theta) = \frac{2\sin\theta\cos\theta}{\cos^2\theta - \sin^2\theta}. \] Divide numerator and denominator by \(\cos^2\theta\): \[ \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}. \]
â Formula: \(\displaystyle \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\), provided \(\tan\theta \neq \pm 1\).
\[ \sin(2\theta) = 2\sin\theta\cos\theta = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25}. \]
First find \(\sin\theta\): \(\sin^2\theta = 1 - \cos^2\theta = 1 - \frac{25}{169} = \frac{144}{169}\), so \(\sin\theta = \frac{12}{13}\). \[ \cos(2\theta) = \cos^2\theta - \sin^2\theta = \frac{25}{169} - \frac{144}{169} = -\frac{119}{169}. \]
\[ \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta} = \frac{2(2)}{1 - (2)^2} = \frac{4}{-3} = -\frac{4}{3}. \]