How do you write the polar equation $θ = \frac{π}{3}$ $theta=pi/3$ in rectangular form?

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Answer 1 · 2016-11-08T20:21:24

Please see the explanation for steps leading to the equation:
$y = ({tan}^{- 1} (\frac{π}{3})) x$ $y = (tan^-1(pi/3))x$

Substitute $tan (\frac{y}{x})$ $tan(y/x)$ for $θ$ $theta$

$tan (\frac{y}{x}) = \frac{π}{3}$ $tan(y/x) = pi/3$

Obtain $\frac{y}{x}$ $y/x$ on the left by using the inverse tangent on both sides:

${tan}^{- 1} (tan (\frac{y}{x})) = {tan}^{- 1} (\frac{π}{3})$ $tan^-1(tan(y/x)) = tan^-1(pi/3)$

$\frac{y}{x} = {tan}^{- 1} (\frac{π}{3})$ $y/x = tan^-1(pi/3)$

Multiply both sides by x:

$y = ({tan}^{- 1} (\frac{π}{3})) x$ $y = (tan^-1(pi/3))x$

Answer 2 · 2016-11-09T09:31:33

$y = \sqrt{3} x$ $y =sqrt3x$

The relation between polar coordinates $(r, θ)$ $(r,theta)$ and Cartesian rectangular coordinates $(x, y)$ $(x,y)$ is given by

$x = r cos θ$ $x=rcostheta$ , $y = r sin θ$ $y=rsintheta$ and $tan θ = \frac{y}{x}$ $tantheta=y/x$

As $θ = \frac{π}{3}$ $theta=pi/3$ , we have $tan θ = \sqrt{3}$ $tantheta=sqrt3$

and equation is

$\frac{y}{x} = \sqrt{3}$ $y/x=sqrt3$

Multiply both sides by $x$ $x$

$y = \sqrt{3} x$ $y =sqrt3x$

How do you write the polar equation theta=pi/3θ=π3theta=pi/3 in rectangular form?