What is the distance between the following polar coordinates?: # (7,(5pi)/4), (2,(9pi)/8) #

Answer 1

#s~=5,27#

#r_1=7# #r_2=2# #theta_1=(5pi)/4# #theta_2=(9pi)/8# #theta_2-theta_1=(9pi)/8-(5pi)/4=(9pi-10pi)/8=-pi/8#
#cos(-pi/8)=0,9#
#s=sqrt(r_1^2+r_2^2-2*r_1*r_2*cos(theta_2-theta_1))#
#s=sqrt(7^2+2^2-2*7*2*0,9)#
#s=sqrt(49+4-28*0,9)#
#s=sqrt(53-25.2)#
#s=sqrt(27,8)# #s~=5.27#
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Answer 2

#P_1P_2 = sqrt(53-28cos((pi)/8)) ~~5.209#

#P_1P_2 = sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_2-theta_1))# #r_1 = 7, theta_1 =(5pi)/4;r_2 =2, theta_2 =(9pi)/8# #P_1P_2 = sqrt(7^2+2^2-2*7*2cos((9pi)/8-(5pi)/4))# #P_1P_2 = sqrt(49+4-28cos(-(pi)/8)# #P_1P_2 = sqrt(53-28cos((pi)/8)) ~~5.209#
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Answer 3

To find the distance between two polar coordinates, we can use the distance formula derived from the Pythagorean theorem. The formula is:

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

Given the coordinates (7, (\frac{5\pi}{4})) and (2, (\frac{9\pi}{8})), let's denote: ( r_1 = 7 ), ( r_2 = 2 ), ( \theta_1 = \frac{5\pi}{4} ), ( \theta_2 = \frac{9\pi}{8} ).

Now, we can plug these values into the distance formula:

[ \text{Distance} = \sqrt{(2^2 + 7^2 - 2 \cdot 2 \cdot 7 \cdot \cos(\frac{9\pi}{8} - \frac{5\pi}{4}))} ]

[ \text{Distance} = \sqrt{(4 + 49 - 28 \cos(\frac{9\pi}{8} - \frac{5\pi}{4}))} ]

[ \text{Distance} = \sqrt{(53 - 28 \cos(\frac{9\pi}{8} - \frac{5\pi}{4}))} ]

Now, let's calculate the difference of the angles: [ \frac{9\pi}{8} - \frac{5\pi}{4} = \frac{9\pi}{8} - \frac{10\pi}{8} = -\frac{\pi}{8} ]

[ \text{Distance} = \sqrt{(53 - 28 \cos(-\frac{\pi}{8}))} ]

[ \text{Distance} = \sqrt{(53 - 28 \cos(\frac{\pi}{8}))} ]

Now, we can calculate the cosine of (\frac{\pi}{8}) which is approximately ( \cos(\frac{\pi}{8}) \approx 0.92388 ).

[ \text{Distance} = \sqrt{(53 - 28 \times 0.92388)} ]

[ \text{Distance} = \sqrt{(53 - 25.85344)} ]

[ \text{Distance} = \sqrt{(27.14656)} ]

[ \text{Distance} \approx 5.21 ]

So, the distance between the polar coordinates is approximately 5.21 units.

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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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