Problem 17 - Essential Calculus

Question

Use the angle feature of a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. $$ \left(\frac{5}{2}, \frac{4}{3}\right) $$

Step-by-Step Solution

Answer

The polar coordinates of the given point \((\frac{5}{2}, \frac{4}{3})\) are \((r,\theta)\), where r and \(\theta\) values are computed in previous steps.

Step 1: Find the radius (r)

Calculate the radius r by using the Pythagorean theorem. This will be the distance from the origin to the point. The formula for r is \(r = \sqrt{x^2 + y^2}\). Substituting \(x = \frac{5}{2}\) and \(y = \frac{4}{3}\), we find that \(r = \sqrt{\left(\frac{5}{2}\right)^2 + \left(\frac{4}{3}\right)^2}\).

Step 2: Calculate the Angle (\(\theta\))

Find \(\theta\) by using the inverse tangent function. This is the angle between the positive x-axis and the line connecting the point with origin. The formula is \(\theta = \arctan(y/x)\). Substituting \(x = \frac{5}{2}\) and \(y = \frac{4}{3}\), we get \(\theta = \arctan(\frac{\frac{4}{3}}{\frac{5}{2}})\). Some graphing utilities may require the angle in degrees rather than radians, so a conversion may be necessary.

Step 3: Generate the Polar Coordinates

Now, use r the radius and \(\theta\) the angle to write the point in polar coordinates. We take the values of r and \(\theta\) as calculated in previous steps and write it as a set of polar coordinates, that is: \( (r,\theta)\) .