Trigonometric addition formulae

Summary

Trigonometric addition formulae help in simplifying and solving equations involving angles. They include addition and double-angle formulae, which are essential for proving identities and solving trigonometric equations.

• Addition Formula — Used to find the sine, cosine, or tangent of the sum or difference of two angles. Example: $\sin(A + B) = \sin A \cos B + \cos A \sin B$
• Double-Angle Formula — Used to express trigonometric functions of double angles in terms of single angles. Example: $\cos(2A) = \cos^2 A - \sin^2 A$
• Simplifying Expressions — Involves converting expressions like $a \sin \theta + b \cos \theta$ into a single trigonometric function. Example: $a \sin \theta + b \cos \theta = R \sin(\theta + \alpha)$ where $R = \sqrt{a^2 + b^2}$

Exam Tips

Key Definitions to Remember

• Addition formula for sine: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• Double-angle formula for cosine: $\cos(2A) = \cos^2 A - \sin^2 A$

Common Confusions

• Mixing up the signs in addition formulae
• Forgetting to apply the correct identity when simplifying expressions

Typical Exam Questions

• How do you express $\sin x + \sqrt{3} \cos x$ in the form $R \sin(x - \alpha)$? Use $R = \sqrt{a^2 + b^2}$ and find $\alpha$ using trigonometric identities.
• What is the double-angle formula for sine? $\sin(2A) = 2 \sin A \cos A$
• How do you prove a trigonometric identity using addition formulae? Use known identities to transform one side of the equation to match the other.

What Examiners Usually Test

• Ability to apply addition and double-angle formulae correctly
• Simplifying expressions involving trigonometric functions
• Proving trigonometric identities using appropriate formulae