What is the distance between the following polar coordinates?: $(4, \frac{- 7 π}{12}), (2, \frac{π}{8})$ $(4,(-7pi)/12), (2,(pi)/8)$

Question

1652 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-07-01T16:07:41

$D \approx 5.4535$ $D~~5.4535$

The distance formula for polar coordinates can be derived from the distance formula for rectangular coordinates

$D = \sqrt{{(x_{2} - x_{1})}_{2}^{2} + {(y_{2} - y_{1})}_{2}^{2}}$ $D=sqrt((x_2-x_1)^2+(y_2-y_1)^2)$

Instead of using $x$ $x$ and $y$ $y$ values, though, we would just plug in their polar equivalents

$x = r cos (θ)$ $x=rcos(theta)$

$y = r sin (θ)$ $y=rsin(theta)$

Plugging those and using a couple of trigonometric identities, you get the following in purely polar coordinates

$D = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} cos (θ_{1} - θ_{2})}$ $D=sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2))$

Plugging in the polar coordinates you have been given, we get

$D = \sqrt{{(4)}^{2} + {(2)}^{2} - 2 (4) (2) cos (- \frac{7 π}{12} - \frac{π}{8})}$ $D=sqrt((4)^2+(2)^2-2(4)(2)cos(-(7pi)/12-pi/8))$

$D = \sqrt{16 + 4 - 16 cos (- \frac{17 π}{24})}$ $D=sqrt(16+4-16cos(-(17pi)/24))$

$D \approx \sqrt{20 - 16 (- 0.60876)}$ $D~~sqrt(20-16(-0.60876))$

$D \approx 5.4535$ $D~~5.4535$

What is the distance between the following polar coordinates?: (4,(-7pi)/12), (2,(pi)/8) (4,−7π12),(2,π8) (4,(-7pi)/12), (2,(pi)/8)