Trigonometry

Need to know

What you need to know

• Use $\cos(2A) = 2\cos^2 A - 1$ when you want to eliminate other trig functions and form a quadratic in $\cos A$.
• Use $\cos(2A) = 1 - 2\sin^2 A$ when you want to form a quadratic in $\sin A$.
• Use $\sin(2A) = 2\sin A \cos A$ to combine or separate terms, often leading to factorisation.

Explanation

Trigonometry Toolkit

1. Start with the fundamental identities $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \sin\theta / \cos\theta$, which arise from the unit circle.
2. Combine two trigonometric terms into one using the harmonic form $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$, where $R = \sqrt{a^2+b^2}$.
3. Simplify expressions involving different angles using the double angle formulae, like $\sin(2\theta) = 2\sin\theta\cos\theta$, derived from compound angle rules.
4. To solve an equation, use identities to simplify, find the principal solution, then use the graph's symmetry to find all other solutions in the required interval.