(4) \( \operatorname{sen}\left(30^{\circ}+\beta\right)-\cos \left(60^{\circ}+p\right)=\sqrt{3} \operatorname{sen} \beta \)

To solve the equation \[ \operatorname{sen}\left(30^{\circ}+\beta\right)-\cos \left(60^{\circ}+p\right)=\sqrt{3} \operatorname{sen} \beta, \] we will use trigonometric identities to simplify the left-hand side. 1. **Expand \(\operatorname{sen}(30^{\circ} + \beta)\)** using the sine addition formula: \[ \operatorname{sen}(30^{\circ} + \beta) = \operatorname{sen}(30^{\circ}) \operatorname{cos}(\beta) + \operatorname{cos}(30^{\circ}) \operatorname{sen}(\beta). \] We know that \(\operatorname{sen}(30^{\circ}) = \frac{1}{2}\) and \(\operatorname{cos}(30^{\circ}) = \frac{\sqrt{3}}{2}\). Thus, \[ \operatorname{sen}(30^{\circ} + \beta) = \frac{1}{2} \operatorname{cos}(\beta) + \frac{\sqrt{3}}{2} \operatorname{sen}(\beta). \] 2. **Expand \(\cos(60^{\circ} + p)\)** using the cosine addition formula: \[ \cos(60^{\circ} + p) = \cos(60^{\circ}) \cos(p) - \sin(60^{\circ}) \sin(p). \] We know that \(\cos(60^{\circ}) = \frac{1}{2}\) and \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\). Thus, \[ \cos(60^{\circ} + p) = \frac{1}{2} \cos(p) - \frac{\sqrt{3}}{2} \sin(p). \] 3. **Substituting back into the equation**: Now we substitute these expansions back into the original equation: \[ \left(\frac{1}{2} \operatorname{cos}(\beta) + \frac{\sqrt{3}}{2} \operatorname{sen}(\beta)\right) - \left(\frac{1}{2} \cos(p) - \frac{\sqrt{3}}{2} \sin(p)\right) = \sqrt{3} \operatorname{sen} \beta. \] 4. **Rearranging the equation**: This simplifies to: \[ \frac{1}{2} \operatorname{cos}(\beta) + \frac{\sqrt{3}}{2} \operatorname{sen}(\beta) - \frac{1}{2} \cos(p) + \frac{\sqrt{3}}{2} \sin(p) = \sqrt{3} \operatorname{sen} \beta. \] 5. **Combining like terms**: Rearranging gives: \[ \frac{1}{2} \operatorname{cos}(\beta) - \frac{1}{2} \cos(p) + \frac{\sqrt{3}}{2} \operatorname{sen}(\beta) - \sqrt{3} \operatorname{sen} \beta + \frac{\sqrt{3}}{2} \sin(p) = 0. \] This can be simplified to: \[ \frac{1}{2} \operatorname{cos}(\beta) - \frac{1}{2} \cos(p) + \left(\frac{\sqrt{3}}{2} - \sqrt{3}\right) \operatorname{sen} \beta + \frac{\sqrt{3}}{2} \sin(p) = 0. \] 6. **Final simplification**: The term \(\frac{\sqrt{3}}{2} - \sqrt{3} = -\frac{\sqrt{3}}{2}\), so we have: \[ \frac{1}{2} \operatorname{cos}(\beta) - \frac{1}{2} \cos(p) - \frac{\sqrt{3}}{2} \operatorname{sen} \beta + \frac{\sqrt{3}}{2} \sin(p) = 0. \] 7. **Isolate terms**: This can be rearranged to isolate \(\operatorname{sen} \beta\) and \(\cos \beta\): \[ \frac{1}{2} \operatorname{cos}(\beta) - \frac{1}{2} \cos(p) + \frac{\sqrt{3}}{2} \sin