What is the distance between $(2, \frac{5 π}{6})$ $(2 ,(5 pi)/6 )$ and $(- 4, \frac{7 π}{4})$ $(-4 , (7 pi )/4 )$ ?

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What is the distance between $(2, \frac{5 π}{6})$ $(2 ,(5 pi)/6 )$ and $(- 4, \frac{7 π}{4})$ $(-4 , (7 pi )/4 )$ ?

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Answer 1 · 2017-03-26T23:37:27

$d = 2 \sqrt{5 - \sqrt{2} - \sqrt{6}} \approx 2.132$ $d = 2 sqrt(5 - sqrt(2) - sqrt(6)) ~~ 2.132$

Explanation:

Polar coordinates: $(r, θ)$ $(r, theta)$

Find location of $(- 4, \frac{7 π}{4})$ $(-4, (7pi)/4)$ :
Start at the x-axis and go $\frac{7 π}{4} = 315^{\circ}$ $(7pi)/4 = 315^@$ , Since $r = - 4$ $r = -4$ , rotate $180^{\circ}$ $180^@$ : $(- 4, \frac{7 π}{4}) = (4, \frac{3 π}{4})$ $(-4, (7pi)/4) = (4, (3pi)/4)$

Convert both to rectangular coordinates $(r cos θ, r sin θ)$ $(r cos theta, r sin theta)$

$(2, \frac{5 π}{6}) = (2 sin (\frac{5 π}{6}), 2 cos (\frac{5 π}{6})) = (- 2 \frac{\sqrt{3}}{2}, 2 \cdot \frac{1}{2}) = (- \sqrt{3}, 1)$ $(2, (5pi)/6) = (2 sin ((5pi)/6), 2 cos ((5pi)/6)) = (-2sqrt(3)/2, 2*1/2) = (-sqrt(3), 1)$

$(4, \frac{3 π}{4}) = (4 cos (\frac{3 π}{4}), 4 sin (\frac{3 π}{4})) = (4 \cdot - \frac{\sqrt{2}}{2}, 4 \cdot \frac{\sqrt{2}}{2}) = (- 2 \sqrt{2}, 2 \sqrt{2})$ $(4, (3pi)/4) = (4 cos ((3pi)/4), 4 sin ((3pi)/4)) = (4*-sqrt(2)/2, 4*sqrt(2)/2) = (-2sqrt(2), 2sqrt(2))$

Find the distance between the points $d = \sqrt{{(y_{2} - y_{1})}_{2}^{2} + {(x_{2} - x_{1})}_{2}^{2}}$ $d = sqrt((y_2-y_1)^2 + (x_2-x_1)^2)$ :

$d = \sqrt{{(2 \sqrt{2} - 1)}^{2} + {(- 2 \sqrt{2} - - \sqrt{3})}^{2}}$ $d = sqrt((2sqrt(2) - 1)^2 + (-2sqrt(2)--sqrt(3))^2)$
$d = \sqrt{(2 \sqrt{2} - 1) (2 \sqrt{2} - 1) + (- 2 \sqrt{2} + \sqrt{3}) (- 2 \sqrt{2} + \sqrt{3})}$ $d = sqrt((2sqrt(2) - 1)(2sqrt(2) - 1) + (-2sqrt(2)+sqrt(3)) (-2sqrt(2)+sqrt(3)))$
$d = \sqrt{9 - 4 \sqrt{2} + 11 - 4 \sqrt{6}}$ $d = sqrt(9-4sqrt(2) + 11 - 4sqrt(6) )$
$d = \sqrt{20 - 4 \sqrt{2} - 4 \sqrt{6}}$ $d = sqrt(20 - 4 sqrt(2) - 4 sqrt(6))$
$d = \sqrt{4 (5 - \sqrt{2} - \sqrt{6})}$ $d = sqrt(4(5 - sqrt(2) - sqrt(6))$
$d = 2 \sqrt{5 - \sqrt{2} - \sqrt{6}} \approx 2.132$ $d = 2 sqrt(5 - sqrt(2) - sqrt(6)) ~~ 2.132$

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What is the distance between $(2, \frac{5 π}{6})$ $(2 ,(5 pi)/6 )$ and $(- 4, \frac{7 π}{4})$ $(-4 , (7 pi )/4 )$ ?

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Explanation:

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What is the distance between $(2, \frac{5 π}{6})$ $(2 ,(5 pi)/6 )$ and $(- 4, \frac{7 π}{4})$ $(-4 , (7 pi )/4 )$ ?