What are the components of the vector between the origin and the polar coordinate $(21, \frac{- π}{8})$ $(21, (-pi)/8)$ ?

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What are the components of the vector between the origin and the polar coordinate $(21, \frac{- π}{8})$ $(21, (-pi)/8)$ ?

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Answer 1 · 2016-11-24T22:16:30

$x = ρ cos θ$ $x = rho cos theta$
$y = ρ sin θ$ $y = rho sin theta$

Explanation:

$x = 21 cos (\frac{π}{8})$ $x = 21 cos (pi/8)$
$y = - 21 sin (\frac{π}{8})$ $y=-21 sin(pi/8)$

$\frac{π}{8}$ $pi/8$ is not an angle for which the value is universally known but you can find it with the trigonometric formula for the bisected angle:

$cos (\frac{π}{8}) = cos (\frac{1}{2} \cdot \frac{π}{4}) = \sqrt{(\frac{1 + cos (\frac{π}{4})}{2})} = \sqrt{(\frac{1 + \sqrt{2}}{2})}$ $cos(pi/8) = cos (1/2*pi/4) = sqrt(((1+cos(pi/4))/2))= sqrt(((1+sqrt(2))/2)$
$sin (\frac{π}{8}) = sin (\frac{1}{2} \cdot \frac{π}{4}) = \sqrt{(\frac{1 - cos (\frac{π}{4})}{2})} = \sqrt{(\frac{1 - \sqrt{2}}{2})}$ $sin(pi/8) = sin (1/2*pi/4) = sqrt(((1-cos(pi/4))/2))= sqrt(((1-sqrt(2))/2)$

and eventually:

$x = 21 \sqrt{(\frac{1 + \sqrt{2}}{2})}$ $x = 21 sqrt(((1+sqrt(2))/2)$
$y = - 21 \sqrt{(\frac{1 - \sqrt{2}}{2})}$ $y=-21sqrt(((1-sqrt(2))/2)$

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What are the components of the vector between the origin and the polar coordinate $(21, \frac{- π}{8})$ $(21, (-pi)/8)$ ?

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Explanation:

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