5.1 If \( 2 \cos 2 \emptyset+\sin 2 \emptyset=R \cos (2 \emptyset-\beta) \) where \( R>0 \) and \( \beta \) is an acute angle. Determine \( R \) and \( \beta \).

To express \( 2 \cos 2\emptyset + \sin 2\emptyset \) in the form \( R \cos (2\emptyset - \beta) \), we start by rewriting the left-hand side: \[ 2 \cos 2\emptyset + \sin 2\emptyset = R \cos (2\emptyset - \beta) = R (\cos 2\emptyset \cos \beta + \sin 2\emptyset \sin \beta) \] We can equate this to \( R \cos (2\emptyset - \beta) \): \[ 2 \cos 2\emptyset + \sin 2\emptyset = R \cos \beta \cos 2\emptyset + R \sin \beta \sin 2\emptyset \] Next, we can group the coefficients of \( \cos 2\emptyset \) and \( \sin 2\emptyset \): \[ (2 - R \cos \beta) \cos 2\emptyset + (1 - R \sin \beta) \sin 2\emptyset = 0 \] Since this must hold for all \( 2\emptyset \), we can set up two equations by equating the coefficients: 1. \( R \cos \beta = 2 \) 2. \( R \sin \beta = 1 \) Now, we can divide these two equations: \[ \frac{R \sin \beta}{R \cos \beta} = \frac{1}{2} \] This simplifies to: \[ \tan \beta = \frac{1}{2} \] To find \( R \), we can use either equation. We can use \( R \sin \beta = 1 \): Using \( \sin \beta \): \[ R \sin \beta = 1 \implies R = \frac{1}{\sin \beta} \] Now, we find \( \sin \beta \) and \( \cos \beta \) using the identity \( \tan \beta = \frac{1}{2} \): From \( \tan \beta = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{2} \), we can think of a right triangle where \( \text{opposite} = 1 \) and \( \text{adjacent} = 2 \). Using the Pythagorean theorem, we find the hypotenuse: \[ \text{hypotenuse} = \sqrt{1^2 + 2^2} = \sqrt{5} \] Thus, \[ \sin \beta = \frac{1}{\sqrt{5}}, \quad \cos \beta = \frac{2}{\sqrt{5}} \] Now substituting \( \sin \beta \) back into the equation for \( R \): \[ R = \frac{1}{\frac{1}{\sqrt{5}}} = \sqrt{5} \] Finally, we check our equations: From \( R \cos \beta \): \[ R \cos \beta = \sqrt{5} \cdot \frac{2}{\sqrt{5}} = 2 \quad \text{(correct)} \] And \( R \sin \beta \): \[ R \sin \beta = \sqrt{5} \cdot \frac{1}{\sqrt{5}} = 1 \quad \text{(correct)} \] Thus, the values of \( R \) and \( \beta \) are: \[ \boxed{R = \sqrt{5}, \beta = \tan^{-1}\left(\frac{1}{2}\right) } \]