What is the distance between the following polar coordinates?: # (21,(4pi)/3), (11,(11pi)/8) #

Answer 1

#10.20 \ "units"# (2dp)

There are two basic methods, which invoke either working in polar coordinates or converting to Cartesian coordinates. The polar coordinate equation is a little harder to remember but easier to use, whereas the rectangular formula is easy to remember (it just relies upon Pythagoras) but invokes more work due to the polar to Cartesian conversion

Polar Coordinates The distance between two polar coordinates #(r_1, theta_1)# and #(r_2, theta_2)# is given by;
#d=sqrt(( r_1^2+ r_2^2 -2r_1r)2 cos(theta_2 - theta_1) )#
So for polar coordinates #(21,(4pi)/3)# and #(11,(11pi)/8)# we have:
#d=sqrt(21^2+11^2-2*21*11cos((11pi)/8-(4pi)/3)# # \ \ =sqrt(441+121-462cos(pi/24))# # \ \ =sqrt(103.9524..)# # \ \ =10.1957...#
Cartesian (Rectangular) Coordinates The distance between two Cartesian coordinates #(x_1, y_1)# and #(x_2, y_2)# is given by:
#d=sqrt((x_1-x_2)^2+(y_1-y_2)^2) #
The Cartesian equivalent of the polar coordinate #(r, theta)# is #(rcos theta, r sin theta)#, So:
#(21,(4pi)/3) rarr (21cos((4pi)/3), 21sin((4pi)/3)) # # " "= (-10.5, -18.1865...)#
#(11,(11pi)/8) rarr (11cos((11pi)/8), 11sin((11pi)/8)) # # " "= (-4.2095..., -10.1626...)#

So s the distance between these coordinates is:

#d=sqrt((-10.5-(-4.2095...))^2 + (-18.1865...-(-10.1626...))^2)# # \ \ =sqrt((-6.2904...)^2 + (-8.0238)^2)# # \ \ =sqrt(39.5701...+64.3823...)# # \ \ =sqrt(103.9524...)# # \ \ =10.1957...#, as above
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Answer 2

To find the distance between two 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 radii, and ( \theta_1 ) and ( \theta_2 ) are the angles.

For the given coordinates: ( r_1 = 21 ), ( \theta_1 = \frac{4\pi}{3} ), ( r_2 = 11 ), ( \theta_2 = \frac{11\pi}{8} )

Substituting the values into the formula:

[ d = \sqrt{(21)^2 + (11)^2 - 2(21)(11)\cos\left(\frac{11\pi}{8} - \frac{4\pi}{3}\right)} ]

[ d = \sqrt{441 + 121 - 462\cos\left(\frac{11\pi}{8} - \frac{4\pi}{3}\right)} ]

[ d = \sqrt{562 - 462\cos\left(\frac{33\pi}{24} - \frac{32\pi}{24}\right)} ]

[ d = \sqrt{562 - 462\cos\left(\frac{\pi}{24}\right)} ]

[ d \approx \sqrt{562 - 462 \times 0.259}]

[ d \approx \sqrt{562 - 119.838}]

[ d \approx \sqrt{442.162}]

[ d \approx 21.029 ]

So, the distance between the given polar coordinates is approximately ( 21.029 ).

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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.

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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.

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