Solved: "The rectangular coordinates of a point are given..."

College
Calculus and Analysis
Integral Calculus
Parametric equations, polar coordinates, and vector-valued functions

Ernstfalld

Answered question

2020-12-22

The rectangular coordinates of a point are given (2, -2). Find polar coordinates of the point. Express theta in radians.

Answer & Explanation

yunitsiL

Skilled2020-12-23Added 108 answers

Let the given rectangular coordinates be (x, y). Then, $(x, y) = (2, - 2)$ Let polar coordinates be $= (r, θ)$ Then using formulas - $r = \sqrt{x^{2} + y^{2}}$
$θ = \tan^{- 1} (\frac{y}{x})$ Using equation (1) - $r = \sqrt{x^{2} + y^{2}}$
$r = \sqrt{(2)^{2} + (- 2)^{2}}$
$r = \sqrt{4 + 4}$
$r = \sqrt{8}$
$r = 2 \sqrt{2}$ Now $θ = \tan^{- 1} (\frac{y}{x})$
$θ = \tan^{- 1} (\frac{- 2}{2})$
$θ = \tan^{- 1} (- 1)$
$θ = \tan^{- 1} (1) [\tan^{- 1} (- x) = - \tan^{- 1} (x)]$
$θ = \tan^{- 1} (\tan π / 4)$
$θ = - π / 4$ Hence, Polar coordinates = $(r, θ) = (2 \sqrt{2}, - \frac{π}{4})$

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