Problem 8 - Essential Calculus

Question

Use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. $$ (8.25,1.3) $$

Step-by-Step Solution

Answer

The polar coordinates (8.25,1.3) convert to the rectangular coordinates approximately (2.2555, 7.8473).

Step 1: Identify the Polar Coordinates

Recognize that the provided values are polar coordinates are denoted as \( (r, \Theta) \), where \( r \) is the radial distance from the origin and \( \Theta \) is the angle from the positive x-axis. Our given coordinates are \( (8.25, 1.3) \).

Step 2: Convert into Rectangular Coordinates

Convert the polar coordinates to rectangular coordinates using the formulas \( x = r*cos(\Theta) \) and \( y = r*sin(\Theta) \). Applying these formulas gives \( x = 8.25*cos(1.3) \approx 2.2555 \) and \( y = 8.25*sin(1.3) \approx 7.8473 \). Thus, the rectangular coordinates are approximately (2.2555, 7.8473).

Step 3: Plot the point

Plot the point on a graph with the origin at (0,0), the x-coordinate as the horizontal distance from the origin, and the y-coordinate as the vertical distance. Our point is in the first quadrant, as both x and y are positive.