Compound Angle Formulae

• Compound Angles: A compound angle is formed by the algebraic sum or difference of two or more individual angles, typically denoted as $(A + B)$ or $(A - B)$. Understanding how to decompose these allows mathematicians to work with complex rotations by breaking them into simpler, known components.

• The Addition Formulae: These are a set of six primary identities (two each for sine, cosine, and tangent) that provide a bridge between the trigonometric ratio of a combined angle and the ratios of its constituent parts. They are not linear operators, meaning $\sin(A+B)$ is never simply $\sin A + \sin B$.

• Foundational Role: These formulae serve as the parent identities from which many other trigonometric rules, such as double-angle and half-angle formulae, are derived. They are indispensable when dealing with wave interference, phase shifts, and coordinate geometry rotations.

• Sine Expansion: The sine of a compound angle is expressed as $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$. Notice that the sine formula 'mixes' the functions (sine and cosine) and 'preserves' the operator sign from the left-hand side to the right-hand side.

• Cosine Expansion: The cosine of a compound angle is expressed as $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$. Unlike sine, the cosine formula 'groups' the functions (cosines together, sines together) and 'reverses' the operator sign, making it a common area for student errors.

• Symmetry and Logic: These patterns emerge from the geometric properties of rotation. The cosine formula's sign flip is a result of the Pythagorean relationship and the way horizontal projections interact during additive rotation.

• Tangent Expansion: The tangent formulae are $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$. In these identities, the numerator sign matches the compound angle's sign, while the denominator sign is always the opposite.

• Conceptual Derivation: The tangent identity is derived by taking the ratio of the sine expansion to the cosine expansion: $\frac{\sin(A+B)}{\cos(A+B)}$. By dividing every term in the resulting fraction by $\cos A \cos B$, the expression simplifies into terms involving only $\tan A$ and $\tan B$.

• Application Limits: These formulae are valid only when the individual tangents ($\tan A, \tan B$) and the compound tangent ($\tan(A \pm B)$) are defined, meaning the angles cannot be odd multiples of $90^\circ$ ($\frac{\pi}{2}$ radians).

• Sign Conventions: A critical distinction between the three functions is how they handle the plus/minus sign. Sine is 'consistent', Cosine is 'contrary', and Tangent is 'split' between its numerator and denominator.

• Term Grouping: Recognizing the structure of the right-hand side is vital for 'reverse' problems. If you see mixed terms ($\sin \cos$), you are looking at a sine expansion; if you see grouped terms ($\cos \cos$), it is a cosine expansion.

• Exact Value Recognition: Exams frequently ask for the 'exact value' of angles like $15^\circ$ or $75^\circ$. Always look to decompose these into standard angles: $15^\circ = 45^\circ - 30^\circ$ and $75^\circ = 45^\circ + 30^\circ$.

• Working Backwards: Be prepared to simplify a long expression into a single trigonometric term. For example, if presented with $\cos 40 \cos 10 + \sin 40 \sin 10$, you should immediately recognize this as the expansion of $\cos(40 - 10) = \cos 30$.

• Verification via Substitution: If you are unsure of a sign in a proof, substitute simple values (like $A=30^\circ, B=60^\circ$) into both sides of your identity. If the numerical values don't match, you likely have a sign error in your expansion.

Memorization Tip: Remember 'CO-sine is CO-ntrary' to help recall that the sign in the cosine expansion is always the opposite of the sign in the bracket.