🧮 Double Angle Formulae
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.
1. Sine Double Angle
Starting from \(\sin(A+B) = \sin A \cos B + \cos A \sin B\),
if we let \( A = B = \theta \), then
\[
\sin(2\theta) = \sin(\theta + \theta)
= \sin\theta\cos\theta + \cos\theta\sin\theta
= 2\sin\theta\cos\theta.
\]
✅ Formula: \(\displaystyle \sin(2\theta) = 2\sin\theta\cos\theta\).
2. Cosine Double Angle
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
\[
\cos(2\theta) = 2\cos^2\theta - 1.
\]
✅ Formulas:
\[
\cos(2\theta) = \cos^2\theta - \sin^2\theta
= 1 - 2\sin^2\theta
= 2\cos^2\theta - 1.
\]
3. Tangent Double Angle
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 \(\cos^2\theta\):
\[
\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}.
\]
✅ Formula:
\(\displaystyle \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\),
provided \(\tan\theta \neq \pm 1\).
📝 Examples
Example 1: If \(\sin\theta = \frac{3}{5}\) and \(\cos\theta = \frac{4}{5}\), find \(\sin(2\theta)\).
\[
\sin(2\theta) = 2\sin\theta\cos\theta
= 2 \times \frac{3}{5} \times \frac{4}{5}
= \frac{24}{25}.
\]
Example 2: If \(\cos\theta = \frac{5}{13}\), find \(\cos(2\theta)\).
First find \(\sin\theta\):
\(\sin^2\theta = 1 - \cos^2\theta = 1 - \frac{25}{169} = \frac{144}{169}\),
so \(\sin\theta = \frac{12}{13}\).
\[
\cos(2\theta) = \cos^2\theta - \sin^2\theta
= \frac{25}{169} - \frac{144}{169}
= -\frac{119}{169}.
\]
Example 3: If \(\tan\theta = 2\), find \(\tan(2\theta)\).
\[
\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}
= \frac{2(2)}{1 - (2)^2}
= \frac{4}{-3}
= -\frac{4}{3}.
\]
📌 Key Points:
- Double angle formulae are derived from sum of angles identities.
- There are three equivalent forms of the cosine double angle formula — choose the one best suited to the problem.
- Watch the domain: \(\tan(2\theta)\) is undefined when \(\tan\theta = \pm 1\).
- These identities are particularly useful in integration, differentiation, and solving trigonometric equations.