1. What Are Compound Angles?
A compound angle is an angle formed by adding or subtracting two angles, typically written as:
\( A + B \quad \text{or} \quad A - B \)
Trigonometric ratios of such angles have beautifully structured formulas that help simplify expressions and evaluate values without a calculator.
2. Formula for sin(A ± B)
The sine compound angle identities express the sine of a sum or difference of two angles in terms of individual sines and cosines:
\( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
\( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
2.1. Example
Find \( \sin(75^\circ) \).
Write 75° as 45° + 30°.
\( \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \)
\( = \dfrac{1}{\sqrt{2}} \cdot \dfrac{\sqrt{3}}{2} + \dfrac{1}{\sqrt{2}} \cdot \dfrac{1}{2} \)
\( = \dfrac{\sqrt{3}+1}{2\sqrt{2}} \)
3. Formula for cos(A ± B)
The cosine identities are similar, but the signs change differently:
\( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
\( \cos(A - B) = \cos A \cos B + \sin A \sin B \)
3.1. Example
Find \( \cos(15^\circ) \).
Write 15° as 45° − 30°.
\( \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ \)
\( = \dfrac{1}{\sqrt{2}} \cdot \dfrac{\sqrt{3}}{2} + \dfrac{1}{\sqrt{2}} \cdot \dfrac{1}{2} \)
\( = \dfrac{\sqrt{3}+1}{2\sqrt{2}} \)
4. Formula for tan(A ± B)
The tangent compound identity arises from the sine and cosine identities:
\( \tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B} \)
\( \tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B} \)
4.1. Example
Find \( \tan(75^\circ) \).
75° = 45° + 30°.
\( \tan 75^\circ = \dfrac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} \)
\( = \dfrac{1 + \dfrac{1}{\sqrt{3}}}{1 - \dfrac{1}{\sqrt{3}}} \)
\( = \dfrac{\sqrt{3}+1}{\sqrt{3}-1} \)
5. Useful Forms for Negative Angles
Compound identities also help simplify expressions involving negative angles:
\( \sin(-\theta) = -\sin \theta \)
\( \cos(-\theta) = \cos \theta \)
\( \tan(-\theta) = -\tan \theta \)
6. Evaluating Non-standard Angles Using Compound Identities
Compound identities allow you to evaluate angles like 15°, 75°, 105°, and more by breaking them into known angles (30°, 45°, 60°).
6.1. Example
Compute \( \sin 105^\circ \).
105° = 60° + 45°.
\( \sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ \)
\( = \dfrac{\sqrt{3}}{2} \cdot \dfrac{1}{\sqrt{2}} + \dfrac{1}{2} \cdot \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{3}+1}{2\sqrt{2}} \)