the equation of straight line in the form of \( x \cos \alpha+y \sin \alpha=p \).

The equation of a straight line in the form \( x \cos \alpha + y \sin \alpha = p \) represents a line in a two-dimensional Cartesian coordinate system. Here, \( \alpha \) is an angle, and \( p \) is a constant that represents the perpendicular distance from the origin to the line. ### Breakdown of the Equation: 1. **\( \cos \alpha \)** and **\( \sin \alpha \)**: - These terms represent the direction cosines of the angle \( \alpha \). The angle \( \alpha \) is measured from the positive x-axis. - \( \cos \alpha \) gives the x-component of the direction, while \( \sin \alpha \) gives the y-component. 2. **\( p \)**: - The constant \( p \) indicates how far the line is from the origin, measured along the perpendicular dropped from the origin to the line. ### Interpretation: - The line described by this equation is perpendicular to the vector \( (\cos \alpha, \sin \alpha) \) and is at a distance \( p \) from the origin. - If \( p \) is positive, the line is on one side of the origin; if \( p \) is negative, the line is on the opposite side. ### Example: For example, if \( \alpha = 30^\circ \) and \( p = 5 \): - \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) - \( \sin 30^\circ = \frac{1}{2} \) The equation of the line would be: \[ x \cdot \frac{\sqrt{3}}{2} + y \cdot \frac{1}{2} = 5 \] This can be rearranged to standard linear form if needed. ### Conclusion: The equation \( x \cos \alpha + y \sin \alpha = p \) is a useful representation of a line in terms of its angle and distance from the origin, providing a geometric interpretation of the line's position in the coordinate plane.