Problem 5 - 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. $$ (5,3 \pi / 4) $$

Step-by-Step Solution

Answer

The rectangular coordinates equivalent to polar coordinates (5, \(3\pi / 4\)) are approximately (-3.54, 3.54).

Step 1: Conversion from polar to rectangular coordinates

Here \((5, 3\pi / 4)\) are polar coordinates (r, θ), which means the distance from origin r is 5 units and the angle θ from the positive x-axis is 3π/4 radian. We can convert r and θ to x and y using the conversion formulas \(x = r \cos(θ)\) and \(y = r \sin(θ)\).

Step 2: Calculate x-coordinate

The rectangular x-coordinate can be calculated using \(x = r \cos(θ)\). For r = 5, θ = 3π/4, we have \(x = 5 \cos(3 \pi / 4)\). You can use your graphing calculator’s cosine function to evaluate this.

Step 3: Calculate y-coordinate

The rectangular y-coordinate can be calculated using \(y = r \sin(θ)\). For r = 5, θ = 3π/4, we have \(y = 5 \sin(3 \pi / 4)\). Use your graphing calculator’s sine function to evaluate this.

Step 4: Plot the point

After evaluating the x and y coordinates, plot this point on your graph.