# What is the distance between the following polar coordinates?: # (7,(3pi)/4), (3,(pi)/8) #

You can use the Law of Cosines.

If you plot the polar coordinates and draw lines from the origin to the points, you will have 2 sides of a triangle. The 3rd side will be the distance between the 2 points.

You can also solve this problem by converting the polar coordinates to rectangular coordinates and using the distance formula.

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To find the distance between two polar coordinates ( (r_1, \theta_1) ) and ( (r_2, \theta_2) ), you can use the formula:

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

Plugging in the values ( r_1 = 7 ), ( \theta_1 = \frac{3\pi}{4} ), ( r_2 = 3 ), and ( \theta_2 = \frac{\pi}{8} ), we get:

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

[ \text{Distance} = \sqrt{49 + 9 - 42\cos\left(\frac{\pi}{8} - \frac{3\pi}{4}\right)} ]

[ \text{Distance} = \sqrt{58 - 42\cos\left(-\frac{5\pi}{8}\right)} ]

[ \text{Distance} = \sqrt{58 - 42\cos\left(\frac{3\pi}{8}\right)} ]

[ \text{Distance} = \sqrt{58 - 42\left(\frac{\sqrt{2 + \sqrt{2}}}{2}\right)} ]

[ \text{Distance} \approx \sqrt{58 - 42\left(\frac{\sqrt{2 + \sqrt{2}}}{2}\right)} ]

[ \text{Distance} \approx \sqrt{58 - 21\sqrt{2 + \sqrt{2}}} ]

[ \text{Distance} \approx \sqrt{58 - 21\sqrt{2 + \sqrt{2}}} ]

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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.

- What is the polar form of #( 23,3 )#?
- How do you find the points of intersection of #r=3+sintheta, r=2csctheta#?
- What is the Cartesian form of #( 3 , (-9pi)/4 ) #?
- What is the area enclosed by #r=cos(4theta-(7pi)/4)+sin(theta+(pi)/8) # between #theta in [pi/3,(5pi)/3]#?
- What is the polar form of #( -34,99 )#?

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