What is the Cartesian form of $(7, \frac{23 π}{3})$ $( 7 , (23pi)/3 )$ ?

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What is the Cartesian form of $(7, \frac{23 π}{3})$ $( 7 , (23pi)/3 )$ ?

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Answer 1 · 2016-02-05T04:56:45

$(\frac{7}{2}, - \frac{7 \sqrt{3}}{2})$ $(7/2, -(7sqrt(3))/2)$

Explanation:

To find cartesian form of $(r, θ)$ $(r,theta)$ we use the formula

$x = r cos (θ)$ $x=rcos(theta)$ and $y = r sin (θ)$ $y=rsin(theta)$

We are given the $(7, \frac{23 π}{3})$ $(7, (23pi)/3)$

$r = 7$ $r=7$ and $θ = \frac{23 π}{3}$ $theta=(23pi)/3$

Let us simplify this $\frac{23 π}{3}$ $(23pi)/3$ into something which is easier to handle.
$\frac{23 π}{3} + \frac{π}{3} = \frac{24 π}{3}$ $(23pi)/3 + pi/3 = (24pi)/3$
$\frac{23 π}{3} + \frac{π}{3} = 8 π$ $(23pi)/3 + pi/3 = 8pi$
$\frac{23 π}{3} = 8 π - \frac{π}{3}$ $(23pi)/3 = 8pi-pi/3$

Also note $cos (2 n π - θ) = cos (θ)$ $cos(2npi-theta) = cos(theta)$
and $sin (2 n π - θ) = - sin (θ)$ $sin(2npi-theta) =-sin(theta)$

$x = r cos (θ)$ $x=rcos(theta)$
$x = 7 cos (8 π - \frac{π}{3})$ $x=7cos(8pi-pi/3)$
$x = 7 cos (\frac{π}{3})$ $x=7cos(pi/3)$
$x = 7 (\frac{1}{2})$ $x=7(1/2)$
$x = \frac{7}{2}$ $x=7/2$

$y = r sin (θ)$ $y=rsin(theta)$
$y = 7 sin (8 π - \frac{π}{3})$ $y=7sin(8pi-pi/3)$
$y = 7 (- sin (\frac{π}{3}))$ $y=7(-sin(pi/3))$
$y = - 7 (\frac{\sqrt{3}}{2})$ $y=-7(sqrt(3)/2)$
$y = - \frac{7 \sqrt{3}}{2}$ $y=-(7sqrt(3))/2$

$(\frac{7}{2}, - \frac{7 \sqrt{3}}{2})$ $(7/2, -(7sqrt(3))/2)$

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What is the Cartesian form of $(7, \frac{23 π}{3})$ $( 7 , (23pi)/3 )$ ?

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Explanation:

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What is the Cartesian form of (7,23π3)( 7 , (23pi)/3 ) ?

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What is the Cartesian form of $(7, \frac{23 π}{3})$ $( 7 , (23pi)/3 )$ ?