Problem 16 - 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. $$ (3 \sqrt{2}, 3 \sqrt{2}) $$

Step-by-Step Solution

Answer

The polar coordinates for the given rectangular coordinates (3√2, 3√2) are (6, π/4).

Step 1: Calculate r

The distance from the origin to the point (x, y) in polar coordinates is called r. This can be calculated by \[ r = \sqrt{x^2 + y^2} \]. So, substitute x=3√2 and y=3√2 to get \[ r= \sqrt{(3\sqrt{2})^2 + (3\sqrt{2})^2} \].

Step 2: Simplify r

Simplify the expression to calcualte the value of r: \[ r = \sqrt{18 + 18} = \sqrt{36} = 6 \].

Step 3: Calculate θ

Now to calculate the angle θ, use the formula \[ θ = \arctan(\frac{y}{x}) \]. Here, \(\frac{y}{x}=1\) since \(y=x=3\sqrt{2}\). Hence calculate \[ θ=\arctan(1) \].

Step 4: Simplify θ

On simplifying, we obtain θ=π/4.

Step 5: Correct the quadrant

Since we are in the first quadrant, there is no need to correct the value of θ obtained. So, the original expression for θ=π/4 is correct.