Find the distance between the points with polar coordinates $\left(2, \frac{\pi}{3}\right)$ and $\left(8, \frac{2\pi}{3}\right)$.

Neetesh Kumar | January 3, 2025

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This is the solution to Math 1c
Assignment: 10.3 Question Number 17
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Step-by-step solution:

Step 1: Distance formula in polar coordinates

The distance $d$ between two points in polar coordinates $(r_1, \theta_1)$ and $(r_2, \theta_2)$ is given by:

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

Here, we have:

• $r_1 = 2$, $\theta_1 = \frac{\pi}{3}$
• $r_2 = 8$, $\theta_2 = \frac{2\pi}{3}$

Step 2: Compute $\cos(\theta_2 - \theta_1)$

The difference between the angles is:

$\theta_2 - \theta_1 = \frac{2\pi}{3} - \frac{\pi}{3} = \frac{\pi}{3}$

So:

$\cos(\theta_2 - \theta_1) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

Step 3: Substitute values into the formula

Substitute $r_1 = 2$, $r_2 = 8$, and $\cos(\theta_2 - \theta_1) = \frac{1}{2}$ into the formula:

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

$d = \sqrt{2^2 + 8^2 - 2(2)(8)\left(\frac{1}{2}\right)}$

Simplify step-by-step:

1. Compute $r_1^2$ and $r_2^2$:

   $2^2 = 4, \quad 8^2 = 64$

2. Compute $2r_1r_2\cos(\theta_2 - \theta_1)$:

   $2(2)(8)\left(\frac{1}{2}\right) = 16$

3. Substitute back:

   $d = \sqrt{4 + 64 - 16}$

4. Simplify:

   $d = \sqrt{52}$

Final Answer:

The distance between the points is:

$d = \boxed{\sqrt{52}}$

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