Trigonometry – Compound angle formulae

Compound Angle Formulae

Quick Reference

\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \] \[ \sin(A - B) = \sin A \cos B - \cos A \sin B \] \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \] \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]

Compound angle formulae allow us to simplify or expand trigonometric expressions involving sums or differences of two angles. These identities are particularly useful in solving trigonometric equations, proving identities, and simplifying expressions.

1. Sine of a Compound Angle

The sine of the sum or difference of two angles is given by: \[ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \] - The sign in the formula matches the sign between \(A\) and \(B\). - Example: \(\sin(60^\circ + 30^\circ) = \sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ\).

2. Cosine of a Compound Angle

The cosine of the sum or difference of two angles is given by: \[ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \] - Notice the opposite sign: for \(\cos(A + B)\), we use a minus; for \(\cos(A - B)\), we use a plus. - Example: \(\cos(60^\circ - 30^\circ) = \cos 60^\circ \cos 30^\circ + \sin 60^\circ \sin 30^\circ\).

3. Tangent of a Compound Angle

The tangent of the sum or difference of two angles is given by: \[ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \] - The signs in the numerator and denominator are opposite. - Example: \(\displaystyle \tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}\).

4. Notes and Tips

• These formulas are valid for all angles measured in degrees or radians.
• Be careful with the signs — especially in the cosine and tangent formulas.
• They are often used in simplifying expressions, solving equations, and deriving other trigonometric identities.
• For \(B = A\), these formulas lead directly to the double angle identities.

Example 1 — Exact values with special angles

Find \(\sin 75^\circ\), \(\cos 15^\circ\), and \(\tan 75^\circ\) exactly.

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\(\displaystyle \sin 75^\circ=\sin(45^\circ+30^\circ)=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ\) \[ =\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\cdot\frac12 =\frac{\sqrt6+\sqrt2}{4}. \]

\(\displaystyle \cos 15^\circ=\cos(45^\circ-30^\circ)=\cos45^\circ\cos30^\circ+\sin45^\circ\sin30^\circ\) \[ =\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\cdot\frac12 =\frac{\sqrt6+\sqrt2}{4}. \]

\(\displaystyle \tan 75^\circ=\tan(45^\circ+30^\circ) =\frac{\tan45^\circ+\tan30^\circ}{1-\tan45^\circ\tan30^\circ} =\frac{1+\frac{1}{\sqrt3}}{1-\frac{1}{\sqrt3}} =\frac{\sqrt3+1}{\sqrt3-1} =2+\sqrt3.\)

Example 2 — Solving a shifted cosine equation

Solve \(\cos(\theta+30^\circ)=\tfrac12\) for \(0^\circ\le \theta<360^\circ\).

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\(\cos u=\tfrac12\Rightarrow u=60^\circ\) or \(u=300^\circ\) (mod \(360^\circ\)). Let \(u=\theta+30^\circ\). Then: \[ \theta+30^\circ=60^\circ \ \Rightarrow\ \theta=30^\circ;\qquad \theta+30^\circ=300^\circ \ \Rightarrow\ \theta=270^\circ. \] Hence \(\boxed{\theta=30^\circ,\,270^\circ}\).

Example 3 — Simplify using \(\sin(A+B)\)

Evaluate \(\sin 50^\circ\cos 10^\circ+\cos 50^\circ\sin 10^\circ\).

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Using \(\sin(A+B)=\sin A\cos B+\cos A\sin B\) with \(A=50^\circ,\,B=10^\circ\), \[ \sin 50^\circ\cos 10^\circ+\cos 50^\circ\sin 10^\circ=\sin(60^\circ)=\frac{\sqrt3}{2}. \]

Example 4 — Exact values in radians

Find \(\sin\frac{\pi}{12}\) exactly.

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\(\displaystyle \frac{\pi}{12}=15^\circ=45^\circ-30^\circ\). \[ \sin\frac{\pi}{12}=\sin(45^\circ-30^\circ) =\sin45^\circ\cos30^\circ-\cos45^\circ\sin30^\circ =\frac{\sqrt6-\sqrt2}{4}. \]

Example 5 — Tangent of a sum from given tangents

Given \(\tan\alpha=2\) and \(\tan\beta=\tfrac13\), find \(\tan(\alpha+\beta)\).

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\[ \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\,\tan\beta} =\frac{2+\frac13}{1-2\cdot\frac13} =\frac{\frac73}{\frac13}=7. \] (Valid provided \(1-\tan\alpha\tan\beta\neq0\), i.e. \(\alpha+\beta\neq 90^\circ+180^\circ k\).)