Questions: The rectangular coordinates of a point are given. Find polar coordinates (r, θ) of this point with θ expressed in radians. Let r>0 and -2π<θ<2π (3,-3) One possibility for the polar coordinates of this point is.

The rectangular coordinates of a point are given. Find polar coordinates (r, θ) of this point with θ expressed in radians. Let r>0 and -2π<θ<2π
(3,-3)
One possibility for the polar coordinates of this point is.

Transcript text: The rectangular coordinates of a point are given. Find polar coordinates $(r, \theta)$ of this point with $\theta$ expressed in radians. Let $r>0$ and $-2 \pi<\theta<2 \pi$ $(3,-3)$ $\qquad$ One possibility for the polar coordinates of this point is $\square$.

Solution

Solution Steps

To convert rectangular coordinates $(x, y)$ to polar coordinates $(r, \theta)$, we use the following formulas:

$ r = \sqrt{x^2 + y^2} $
$ \theta = \arctan\left(\frac{y}{x}\right) $

Given the point $(3, -3)$:

Calculate $ r $ using the distance formula.
Calculate $ \theta $ using the arctangent function and adjust the angle to ensure it falls within the specified range.

Step 1: Calculate $ r $

To find the polar coordinate $ r $, we use the formula:

\[ r = \sqrt{x^2 + y^2} \]

Substituting the values $ x = 3 $ and $ y = -3 $:

\[ r = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 4.2426 \quad (\text{rounded to four significant digits}) \]

Step 2: Calculate $ \theta $

To find the polar coordinate $ \theta $, we use the formula:

\[ \theta = \arctan\left(\frac{y}{x}\right) \]

Substituting the values $ x = 3 $ and $ y = -3 $:

\[ \theta = \arctan\left(\frac{-3}{3}\right) = \arctan(-1) = -0.7854 \quad (\text{rounded to four significant digits}) \]

Step 3: Ensure $ \theta $ is in the correct range

The calculated angle $ \theta = -0.7854 $ is already within the specified range of $ -2\pi < \theta < 2\pi $.