# [[Amplitude-Phase Form of Sine]] (Mathematics > Trigonometry)
## Definition
The amplitude-phase form of a sine function expresses a trigonometric expression involving both sine and cosine terms as a single sine function with an amplitude and phase shift. Specifically, any expression of the form:
$
a \sin x \pm b \cos x
$
can be rewritten as:
$
R \sin(x \pm \phi)
$
where:
- $R$ is the **amplitude** of the function.
- $\phi$ is the **phase shift** of the function.
Here, $R > 0$ and $\phi \in \left(0, \frac{\pi}{2}\right)$.
## Key Concepts
- **Amplitude** $R$: Represents the peak value of the trigonometric expression.
- **Phase Shift** $\phi$: The angle by which the function is shifted horizontally.
- **Trigonometric Transformation**: Any linear combination of sine and cosine functions with the same argument can be expressed in amplitude-phase form.
- **Angular Frequency** $x$: Represents the variable in terms of which the phase and amplitude are considered.
## Important Properties
1. The amplitude $R$ is found as $R = \sqrt{a^2 + b^2}$.
2. The phase shift $\phi$ is determined by $\tan \phi = \frac{b}{a}$.
3. The expression $a \sin x + b \cos x$ retains the same frequency as $\sin x$ and $\cos x$.
## Essential Formulas
- **Amplitude**:
$
R = \sqrt{a^2 + b^2}
$
- **Phase Shift**:
$
\phi = \tan^{-1} \left(\frac{b}{a}\right)
$
- **Amplitude-Phase Form**:
$
a \sin x \pm b \cos x = R \sin(x \pm \phi)
$
## Core Examples
1. **Basic Example**: Rewrite $3 \sin x + 4 \cos x$ in amplitude-phase form.
- Calculate $R = \sqrt{3^2 + 4^2} = 5$.
- Calculate $\phi = \tan^{-1} \left(\frac{4}{3}\right)$.
- Therefore, $3 \sin x + 4 \cos x = 5 \sin(x + \phi)$ where $\phi = \tan^{-1}\left(\frac{4}{3}\right)$.
2. **Advanced Example**: Rewrite $5 \sin x - 12 \cos x$ in amplitude-phase form.
- Calculate $R = \sqrt{5^2 + (-12)^2} = 13$.
- Calculate $\phi = \tan^{-1} \left(\frac{-12}{5}\right)$.
- Thus, $5 \sin x - 12 \cos x = 13 \sin(x - \phi)$, with $\phi = \tan^{-1}\left(\frac{-12}{5}\right)$.
## Related Theorems/Rules
- **Pythagorean Identity**: Used in calculating $R = \sqrt{a^2 + b^2}$.
- **Tangent Inverse Function**: Used for phase calculation, $\phi = \tan^{-1} \left(\frac{b}{a}\right)$.
## Common Pitfalls
- Confusing $a$ and $b$ in the phase calculation formula $\phi = \tan^{-1} \left(\frac{b}{a}\right)$.
- Misidentifying the amplitude $R$ as $a + b$ instead of $\sqrt{a^2 + b^2}$.
## Related Topics
- [[Sine Function]]
- [[Cosine Function]]
- [[Transformed Sine and Cosine Functions]]
- [[Amplitude]]
- [[Phase Shift]]
- [[Horizontal Translations of Functions]]
- [[Unit Circle Overview]]
- [[Comparison of Sine and Cosine Functions]]
- [[trigonometric_graphs]]
- [[Horizontal Scaling of the Sine Function]]
- [[Oddness of the Sine Function]]
- [[Evenness of the Cosine Function]]
- [[Sum and Difference Formulas for Sine]]
- [[Sum and Difference Formulas for Cosine]]
- [[Double-Angle Formula for Sine]]
- [[Double-Angle Formula for Cosine]]
- [[Trigonometric Identities]]
- [[Harmonic Oscillations]]
- [[Vector Representation of Complex Numbers]]
## Quick Review Questions
1. What is the formula to find the amplitude $R$ in the expression $a \sin x + b \cos x$?
2. How can you determine the phase shift $\phi$ for the expression $a \sin x + b \cos x$?
***
![[Amplitude-Phase_Form_of_Sine_visualization]]