Argument Synthesis Method and Its Proof

Core Concept

Argument synthesis method (also known as the R method) is a technique to convert a linear trigonometric expression of the form

$$a\cos\alpha + b\sin\alpha$$

into a single trigonometric function. The core idea is to construct an appropriate amplitude $R$ and phase angle $\phi$ such that the expression can be written as

$$R\sin(\alpha + \phi) \quad\text{or}\quad R\cos(\alpha - \phi).$$

Derivation Steps

Amplitude Calculation

The amplitude $R$ has the geometric interpretation as the magnitude of the vector $(a, b)$, so

$$R = \sqrt{a^2 + b^2}$$

For example, for $\cos\alpha + \sqrt{3}\sin\alpha$:

$$R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2$$

Determining the Phase Angle

Depending on the target form (sine or cosine), the calculation of $\phi$ differs slightly, and quadrant determination is necessary:

• Sine form: Let

  $$a\cos\alpha + b\sin\alpha = R\sin(\alpha + \phi)$$

  Expanding using the angle sum formula:

  $$R\sin(\alpha + \phi) = R[\sin\alpha\cos\phi + \cos\alpha\sin\phi]$$

  Comparing coefficients gives:

  $$\begin{cases} R\cos\phi = b \\ R\sin\phi = a \end{cases} \quad\Rightarrow\quad \begin{cases} \cos\phi = \dfrac{b}{R} \\ \sin\phi = \dfrac{a}{R} \end{cases}$$

  Therefore,

  $$\phi = \arctan2\bigl(a,\,b\bigr)$$

• Cosine form: Let

  $$a\cos\alpha + b\sin\alpha = R\cos(\alpha - \phi)$$

  Expanding:

  $$R\cos(\alpha - \phi) = R[\cos\alpha\cos\phi + \sin\alpha\sin\phi]$$

  Comparing coefficients:

  $$\begin{cases} R\cos\phi = a \\ R\sin\phi = b \end{cases} \quad\Rightarrow\quad \begin{cases} \cos\phi = \dfrac{a}{R} \\ \sin\phi = \dfrac{b}{R} \end{cases}$$

  Therefore,

  $$\phi = \arctan2\bigl(b,\,a\bigr)$$

Note: If only $\arctan\left(\frac{b}{a}\right)$ is used without considering signs, the quadrant may be misjudged. Use $\arctan2$ or sign analysis.

Proof of Correctness

Using the cosine form as an example:

1. Define

   $$R = \sqrt{a^2 + b^2}, \quad \cos\phi = \frac{a}{R}, \quad \sin\phi = \frac{b}{R}$$

2. Expand the right-hand side:

   $$R\cos(\alpha - \phi) = R\bigl[\cos\alpha\cos\phi + \sin\alpha\sin\phi\bigr] = R\cos\phi \cdot \cos\alpha + R\sin\phi \cdot \sin\alpha$$

3. Substitute the definitions:

   $$R\cos\phi = a, \quad R\sin\phi = b \quad \Longrightarrow \quad a\cos\alpha + b\sin\alpha$$

Thus the identity holds. The sine form can be proved analogously.

Concrete Example

Example: Convert $3\cos x - 4\sin x$ into a single trigonometric function.

1. Calculate amplitude:

   $$R = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5$$

2. Determine phase angle (cosine form):

   $$\cos\phi = \frac{3}{5}, \quad \sin\phi = -\frac{4}{5}$$

   The phase angle $\phi$ is in the fourth quadrant:

   $$\phi = -\,\arctan\left(\tfrac{4}{3}\right)$$

3. Synthesized result:

   $$3\cos x - 4\sin x = 5\cos\left(x + \arctan\tfrac{4}{3}\right)$$

4. Verification:

   $$5\cos\left(x + \arctan\frac{4}{3}\right) = 5 \left[\cos x \cdot \frac{3}{5} - \sin x \cdot \frac{4}{5}\right] = 3\cos x - 4\sin x$$

Important Notes

1. Quadrant Determination
   The phase angle $\phi$ must be determined based on the signs of $a$ and $b$.

2. Form Selection

   • For differentiation, prefer the sine form.
   • For integration, prefer the cosine form.

3. Frequency Consistency
   This method only applies to superpositions of trigonometric functions with the same frequency.

4. Connection to Complex Numbers
   Corresponds to complex multiplication: $a + bi = R e^{i\phi}$, representing a “rotation + scaling” transformation.

Summary

The argument synthesis method converts $a\cos\alpha + b\sin\alpha$ into a single trigonometric function through amplitude $R$ and phase angle $\phi$:

• Compute $R = \sqrt{a^2 + b^2}$
• Determine $\phi$ based on the target form
• Applicable in analytic geometry, signal analysis, and related fields.