Questions: 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). ((-7,7)) One possibility for the polar coordinates of this point is (Simplify your answer. Type an ordered pair. Type your answer in radians. Type exact answers, using (pi) as needed. Use integers or fractions for any numbers in the expression.)

$The rectangular coordinates of a point are given. Find polar coordinates ((r, theta)) of this point with (theta) expressed in radians. Let (r>0) and (-2 pi<theta<2 pi). ((-7,7)) One possibility for the polar coordinates of this point is (Simplify your answer. Type an ordered pair. Type your answer in radians. Type exact answers, using (pi) as needed. Use integers or fractions for any numbers in the expression.)$

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 $-2 \pi<\theta<2 \pi$. \[ (-7,7) \] One possibility for the polar coordinates of this point is $\square$ (Simplify your answer. Type an ordered pair. Type your answer in radians. Type exact answers, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.)

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 $(-7, 7)$, we can calculate $r$ and $\theta$ using these formulas.

Solution Approach

Calculate $r$ using the formula $ r = \sqrt{x^2 + y^2} $.
Calculate $\theta$ using the formula $\theta = \arctan\left(\frac{y}{x}\right)$.
Adjust $\theta$ to ensure it falls within the range $-2\pi < \theta < 2\pi$.

Step 1: Calculate $ r $

To find the polar coordinate $ r $, we use the formula: \[ r = \sqrt{x^2 + y^2} \] Substituting the values $ x = -7 $ and $ y = 7 $: \[ r = \sqrt{(-7)^2 + 7^2} = \sqrt{49 + 49} = \sqrt{98} = 9.8995 \quad (\text{rounded to four significant digits}) \]

Step 2: Calculate $ \theta $

To find the polar coordinate $ \theta $, we use the formula: \[ \theta = \arctan\left(\frac{y}{x}\right) \] Substituting the values: \[ \theta = \arctan\left(\frac{7}{-7}\right) = \arctan(-1) \] Since the point $(-7, 7)$ is in the second quadrant, we adjust $ \theta $: \[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \quad \text{or approximately } 2.3562 \quad (\text{rounded to four significant digits}) \]