# [[Amplitude-Phase Form of Cosine]] (Mathematics > Trigonometry) ## Definition The amplitude-phase form of a cosine function is used to express a trigonometric expression involving both cosine and sine terms as a single cosine function with an amplitude and phase shift. Specifically, any expression of the form: $ a \cos x \pm b \sin x $ can be rewritten as: $ R \cos(x \mp \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** $R$: The maximum value of the trigonometric expression. - **Phase Shift** $\phi$: The horizontal shift in the cosine function. - **Trigonometric Transformation**: A combination of cosine and sine terms with the same argument can be expressed in amplitude-phase form using cosine. - **Angular Frequency** $x$: The argument of the trigonometric function. ## Important Properties 1. The amplitude $R$ is given by $R = \sqrt{a^2 + b^2}$. 2. The phase shift $\phi$ is determined by $\tan \phi = \frac{b}{a}$. 3. Expressions $a \cos x + b \sin x$ and $a \cos x - b \sin x$ both retain the same frequency as $\cos x$ and $\sin x$. ## Essential Formulas - **Amplitude**: $ R = \sqrt{a^2 + b^2} $ - **Phase Shift**: $ \phi = \tan^{-1} \left(\frac{b}{a}\right) $ - **Amplitude-Phase Form (Cosine)**: $ a \cos x \pm b \sin x = R \cos(x \mp \phi) $ ## Core Examples 1. **Basic Example**: Rewrite $3 \cos x + 4 \sin x$ in amplitude-phase form. - Calculate $R = \sqrt{3^2 + 4^2} = 5$. - Calculate $\phi = \tan^{-1} \left(\frac{4}{3}\right)$. - Therefore, $3 \cos x + 4 \sin x = 5 \cos(x - \phi)$ where $\phi = \tan^{-1}\left(\frac{4}{3}\right)$. 2. **Advanced Example**: Rewrite $4 \sin x + \cos x$ in amplitude-phase form. - Calculate $R = \sqrt{1^2 + 4^2} = \sqrt{17}$. - Calculate $\phi = \tan^{-1} \left(\frac{4}{1}\right)$. - Thus, $4 \sin x + \cos x = \sqrt{17} \cos(x - \phi)$, with $\phi = \tan^{-1}\left(4\right)$. ## Related Theorems/Rules - **Pythagorean Identity**: Supports the calculation of $R = \sqrt{a^2 + b^2}$. - **Tangent Inverse Function**: Used for determining the phase shift $\phi$ as $\tan^{-1} \left(\frac{b}{a}\right)$. ## Common Pitfalls - Mixing up $a$ and $b$ in the phase calculation formula $\phi = \tan^{-1} \left(\frac{b}{a}\right)$. - Confusing the form with sine's amplitude-phase form, which uses $R \sin(x \pm \phi)$. ## Related Topics - [[Amplitude]] - [[Horizontal Translations of Functions]] - [[General Transformations of Functions]] - [[Trigonometric Graphs]] - [[Comparison of Sine and Cosine Functions]] - [[Derivatives of Trigonometric Functions]] - [[Horizontal Scaling of the Sine Function]] - [[Transformed Sine and Cosine Functions]] - [[Sum and Difference Formulas for Sine]] - [[Sum and Difference Formulas for Cosine]] - [[Double-Angle Formula for Sine]] - [[Double-Angle Formula for Cosine]] - [[Amplitude-Phase Form of Sine]] - [[Trigonometric Identities]] - [[Polar Form of Complex Numbers]] ## Quick Review Questions 1. How do you calculate the amplitude $R$ for $a \cos x + b \sin x$? 2. What is the phase shift $\phi$ in terms of $a$ and $b$? *** ![[Amplitude-Phase_Form_of_Cosine_visualization]]