What is the distance between the following polar coordinates?: <mstyle displaystyle="true">

Raphael Mccullough 2022-04-12 Answered

What is the distance between the following polar coordinates?:

(1, \frac{3 π}{4}), (5, \frac{5 π}{8})

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Madalynn Acosta

Answered 2022-04-13 Author has 15 answers

Step 1
Call the two points A and B:

A = (r_{a}, θ_{a}) = (1, \frac{3 π}{4})

B = (r_{b}, θ_{b}) = (5, \frac{5 π}{8})

Convert them to rectangular form:

A = (x_{a}, y_{a})

x_{a} = r_{a} \cos (θ_{a}) = 1 \cdot \cos (\frac{3 π}{4}) = - \frac{\sqrt{2}}{2}

y_{a} = r_{a} \sin (θ_{a}) = 1 \cdot \sin (\frac{3 π}{4}) = \frac{\sqrt{2}}{2}

A = (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \approx (- 0.707, 0.707)

B = (x_{b}, y_{b})

x_{b} = r_{b} \cos (θ_{b}) = 5 \cdot \cos (\frac{5 π}{8}) = - 5 \frac{\sqrt{2 - \sqrt{2}}}{2}

y_{b} = r_{b} \sin (θ_{b}) = 5 \cdot \sin (\frac{5 π}{8}) = 5 \frac{\sqrt{2 + \sqrt{2}}}{2}

B = (- 5 \frac{\sqrt{2 - \sqrt{2}}}{2}, 5 \frac{\sqrt{2 + \sqrt{2}}}{2}) \approx (- 1.913, 4.619)

Now apply the Pythagorean Theorem to find the length of the line segment

\bar{A B}

| | \bar{A B} | | = \sqrt{{(x_{b} - x_{a})}^{2} + {(y_{b} - y_{a})}^{2}}

| | \bar{A B} | | \approx \sqrt{{((- 1.913) - (- 0.707))}^{2} + {(4.619 - 0.707)}^{2}}

| | \bar{A B} | | \approx \sqrt{{(- 1.206)}^{2} + {(3.912)}^{2}} = \sqrt{1.455 + 15.306}

| | \bar{A B} | | \approx \sqrt{16.761} \approx 4.094

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