5.6.3 R addition formulae Rcos Rsin etc

The R addition formulae involve expressing a linear combination of sine and cosine functions, such as $a\ \cos \theta + b \sin \theta$, in a single trigonometric form like $R \cos(\theta - \alpha)$ or $R \sin(\theta + \alpha)$. This is particularly useful in solving trigonometric equations and simplifying expressions.

1. Expressing $a\ \cos \theta + b \sin \theta$ as $R\ \cos(\theta \pm \alpha)$

2. Derivation:

Starting with the identity: $R \cos(\theta - \alpha) = R (\cos \theta \cos \alpha + \sin \theta \sin \alpha)$ Expanding this: $R \cos(\theta - \alpha) = R \cos \alpha \cos \theta + R \sin \alpha \sin \theta$ Now, compare this to the original expression $a\ \cos \theta + b \sin \theta$. For the two expressions to be equivalent, we must have: $a = R \cos \alpha \quad \text{and} \quad b = R \sin \alpha$

To find $\ R \ and \ \alpha$:

• Magnitude $\ R$: $R = \sqrt{a^2 + b^2}$
• Angle $\ \alpha$: $\tan \alpha = \frac{b}{a}$ The angle $\alpha$ can be found using the inverse tangent function, $\alpha = \tan^{-1} \left(\frac{b}{a}\right) .$

3. Expressing $a \cos \theta + b \sin \theta$ as$\ R \sin(\theta \pm \alpha)$)

Similarly, the expression can also be rewritten as: $a \cos \theta + b \sin \theta = R \sin(\theta + \alpha)$ Where:

• $R = \sqrt{a^2 + b^2}$
• $\tan \alpha = \frac{a}{b}$ This identity is useful in certain contexts, especially when solving equations where a sine function might be more convenient.

4. Example Problems Using R Addition Formulae:

5. Applications of R Addition Formulae:

• Solving Trigonometric Equations: These formulae simplify the process of solving trigonometric equations by reducing the expression to a single trigonometric function.
• Oscillatory Motion: In physics, particularly in oscillations and wave motion, expressing sums of sine and cosine functions as a single sine or cosine function can simplify analysis.
• AC Circuits: In electrical engineering, these formulae are used to analyse alternating current (AC) circuits where voltages and currents are sinusoidal and may be out of phase.

Summary: