# What is the distance between #(2 , (5 pi)/8 )# and #(3 , (1 pi )/3 )#?

The distance between those two coordinates is

You can use the law of cosines to do that.

Let me illustrate why:

Polar coordinates

#(r, theta)# are defined by the radius#r# and the angle#theta# .Imagine lines leading from the pole to your respective polar coordinates. Those lines represent two sides of a triangle with lengths

#A = 3# and#B = 2# . The distance between those two coordinates being the third side,#C# .

Furthermore, the angle between

#A# and#B# can be computed as the difference between the two angles of your polar coordinates:

#gamma = (5pi)/8 - pi/3 = (7pi)/24# Thus, the length of the side

#C# can be found with the help of law of cosines on that triangle:

#C^2 = A^2 + B^2 - 2AB cos(gamma)#

#= 3^2 + 2^2 - 2 * 3 * 2 * cos((7pi)/24)#

#= 13 - 12 cos((7pi)/24)#

#=> C = sqrt(13 - 12 cos((7pi)/24)) ~~2.39#

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To find the distance between two points ((x_1, y_1)) and ((x_2, y_2)) in a two-dimensional coordinate system, we use the distance formula:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Substituting the given coordinates:

[ d = \sqrt{(3 - 2)^2 + \left(\frac{\pi}{3} - \frac{5\pi}{8}\right)^2} ]

[ d = \sqrt{1^2 + \left(\frac{8\pi - 15\pi}{24}\right)^2} ]

[ d = \sqrt{1 + \left(\frac{-7\pi}{24}\right)^2} ]

[ d = \sqrt{1 + \frac{49\pi^2}{576}} ]

[ d = \sqrt{\frac{576 + 49\pi^2}{576}} ]

[ d = \frac{\sqrt{576 + 49\pi^2}}{24} ]

Therefore, the distance between the points ((2, \frac{5\pi}{8})) and ((3, \frac{\pi}{3})) is ( \frac{\sqrt{576 + 49\pi^2}}{24} ) units.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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