# What is the distance between the following polar coordinates?: # (3,(-15pi)/12), (5,(-3pi)/8) #

Polar coordinates are written as

If we plot

We know the length of two of the sides of the triangle, since the

If the result was larger than

Finally, we know from trigonometry that if we know two sides and the angle in between, we can calculate the length of the other side using the Law of Cosines:

we can use this law to arrive at an equation for the distance between two polar points,

or in this case,

p.s. note the lack of absolute value signs in the final equation. That's because

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To find the distance between the given polar coordinates ((3, \frac{-15\pi}{12})) and ((5, \frac{-3\pi}{8})), you can use the formula for the distance between two points in polar coordinates:

[ \text{Distance} = \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 of the two points respectively.

Substituting the given values, you get:

[ \text{Distance} = \sqrt{3^2 + 5^2 - 2(3)(5) \cos\left(\frac{-3\pi}{8} - \frac{-15\pi}{12}\right)} ]

[ \text{Distance} = \sqrt{9 + 25 - 30 \cos\left(\frac{-3\pi}{8} + \frac{5\pi}{4}\right)} ]

Now, you can calculate the cosine term:

[ \cos\left(\frac{-3\pi}{8} + \frac{5\pi}{4}\right) = \cos\left(\frac{-3\pi}{8} + \frac{10\pi}{8}\right) = \cos\left(\frac{7\pi}{8}\right) ]

Then, use this value to find the distance:

[ \text{Distance} = \sqrt{9 + 25 - 30 \cos\left(\frac{7\pi}{8}\right)} ]

[ \text{Distance} = \sqrt{34 - 30 \cos\left(\frac{7\pi}{8}\right)} ]

You would need to calculate the value of ( \cos\left(\frac{7\pi}{8}\right) ) and then substitute it back into the equation to find the distance.

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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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