What is the distance the polar coordinates #(-2 ,( -3 )/8 )# and #(6 ,(-7 pi )/4 )#?

Answer 1

#=sqrt(40-24sqrt(((sqrt2-1)/(2sqrt2))))5.551#, nearly.

After correcting the first angle as #-3/8pi#, the two points are

P(-2, -3/8pi) or, with positive r, P(2. -3/8pi-pi) =#

# P(2, -11/8pi) and Q(6, -7/4pi)#
Now, in the #triangle POQ#, where O is the pole r = 0,
#OP = 2, OQ = 6 and angle POQ = (--7/4pi-(-11/8pi))=-3/8pi=-67.5^o#.
Use #PQ=sqrt(OP^2+OQ^2-2 OP OQ cos angle POQ#
#=sqrt(4+36-24cos(-67.5^o)#
Here, #cos(-67.5^0)=cos(67.5^o)#
#=sin (90-67.5)^o=sin 22.5^o#
#=sqrt((1-cos 45^0)/2)=sqrt(((sqrt2-1)/(2sqrt2)) #. So,
#PQ=sqrt(40-24sqrt(((sqrt2-1)/(2sqrt2))))=5.551#, nearly -
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Answer 2

To find the distance between two points given in polar coordinates, you can use the formula:

[ d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_2 - \theta_1)} ]

Where:

  • ( r_1 ) and ( r_2 ) are the magnitudes of the polar coordinates
  • ( \theta_1 ) and ( \theta_2 ) are the angles of the polar coordinates

Given the polar coordinates (-2, (-3π)/8) and (6, (-7π)/4):

  • ( r_1 = -2 ) and ( \theta_1 = \frac{-3\pi}{8} )
  • ( r_2 = 6 ) and ( \theta_2 = \frac{-7\pi}{4} )

Plugging these values into the formula:

[ d = \sqrt{(-2)^2 + 6^2 - 2(-2)(6)\cos\left(\frac{-7\pi}{4} - \frac{-3\pi}{8}\right)} ]

[ d = \sqrt{4 + 36 - 2(-12)\cos\left(\frac{-7\pi}{4} + \frac{3\pi}{8}\right)} ]

[ d = \sqrt{40 - 24\cos\left(\frac{-7\pi}{4} + \frac{3\pi}{8}\right)} ]

[ d = \sqrt{40 - 24\cos\left(\frac{-14\pi + 3\pi}{8}\right)} ]

[ d = \sqrt{40 - 24\cos\left(\frac{-11\pi}{8}\right)} ]

[ d = \sqrt{40 - 24\left(\frac{\sqrt{2}}{2}\right)} ]

[ d = \sqrt{40 - 12\sqrt{2}} ]

[ d \approx \sqrt{40} - \sqrt{12}\sqrt{2} ]

[ d \approx 6.3246 - 3.4641 ]

[ d \approx 2.8605 ]

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Answer from HIX Tutor

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.

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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