Compound Angle Identities and Double-Angle Identities

Introduction

Compound angle identities are foundational tools in trigonometry, enabling the simplification of trigonometric functions involving sums or differences of angles. They also provide the basis for deriving double-angle identities, which are widely used in mathematics and physics.

Compound Angle Identities

Sine of Compound Angles

\(\sin(A + B) = \sin A \cos B + \cos A \sin B\)

\(\sin(A – B) = \sin A \cos B – \cos A \sin B\)

These formulas decompose the sine of a sum or difference of two angles into simpler trigonometric components.

Cosine of Compound Angles

\(\cos(A + B) = \cos A \cos B – \sin A \sin B\)

\(\cos(A – B) = \cos A \cos B + \sin A \sin B\)

Cosine formulas express the sum or difference of angles using products of cosine and sine.

Tangent of Compound Angles

\(\tan(A + B) = \frac{\tan A + \tan B}{1 – \tan A \tan B}\)

\(\tan(A – B) = \frac{\tan A – \tan B}{1 + \tan A \tan B}\)

The tangent formulas arise by dividing the sine compound formula by the cosine compound formula.

Double-Angle Identities

Sine Double Angle

\(\sin(2A) = 2\sin A \cos A\)

Derived by applying the sine addition formula: \(\sin(2A) = \sin(A + A)\).

Cosine Double Angle

\(\cos(2A) = \cos^2 A – \sin^2 A\)

Alternate forms derived using the Pythagorean identity:

• \(\cos(2A) = 2\cos^2 A – 1\)
• \(\cos(2A) = 1 – 2\sin^2 A\)

Tangent Double Angle

\(\tan(2A) = \frac{2\tan A}{1 – \tan^2 A}\)

Applicable when \(|\tan A| \neq 1\).

Applications and Examples

Simplify \(\sin(75^\circ)\)

Rewrite \(75^\circ\) as \(45^\circ + 30^\circ\) and use the sine addition formula:

\(\sin(75^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)\)

Substitute values:

\(\sin(75^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\)

Derive \(\cos(2A)\) from \(\cos(A + B)\)

Substitute \(B = A\) in \(\cos(A + B) = \cos A \cos B – \sin A \sin B\):

\(\cos(2A) = \cos^2 A – \sin^2 A\)

Using the Pythagorean identity, alternate forms are derived:

\(\cos(2A) = 2\cos^2 A – 1\) and \(\cos(2A) = 1 – 2\sin^2 A\).

Solve \(\tan(2x) = \sqrt{3}\)

Using the tangent double-angle formula:

\(\tan(2x) = \sqrt{3} \quad \implies \quad 2x = \frac{\pi}{3} + n\pi\)

\(x = \frac{\pi}{6} + \frac{n\pi}{2}, \, n \in \mathbb{Z}\)

Connections

• Applications: Used in calculus, physics, and engineering, e.g., Fourier series, wave motion, and harmonic analysis.
• De Moivre’s Theorem: Expanding \((\cos x + i\sin x)^n\) relies on compound angle identities.