In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (5, 0)

Verified step by step guidance

1. Recall that to convert rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use the formulas: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\).
2. Calculate the radius \(r\) by substituting \(x = 5\) and \(y = 0\) into the formula: \(r = \sqrt{5^2 + 0^2}\).
3. Simplify the expression for \(r\) to find the distance from the origin to the point.
4. Find the angle \(\theta\) by evaluating \(\arctan\left(\frac{0}{5}\right)\), which gives the angle the point makes with the positive \(x\)-axis.
5. Consider the quadrant where the point \((5, 0)\) lies to determine the correct value of \(\theta\) in radians, remembering that if \(x > 0\) and \(y = 0\), \(\theta\) is either \(0\) or \(2\pi\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates (x, y) represent a point's position on a plane using horizontal and vertical distances from the origin. The x-value indicates horizontal displacement, while the y-value indicates vertical displacement. Understanding these coordinates is essential for converting to polar form.

Polar Coordinates

Polar coordinates express a point's location using a distance from the origin (r) and an angle (θ) measured from the positive x-axis. The distance r is always non-negative, and θ is typically given in radians. This system is useful for problems involving rotation or circular motion.

Conversion Between Rectangular and Polar Coordinates

To convert from rectangular (x, y) to polar (r, θ), calculate r = √(x² + y²) and θ = arctangent(y/x). Special attention is needed when x = 0 or when determining the correct quadrant for θ. Expressing θ in radians ensures consistency in trigonometric calculations.