Answered: "What is the distance between the following polar..."

MMDCCVIII446

Answered question

2022-02-04

What is the distance between the following polar coordinates?:

(2, \frac{5 π}{12}), (1, \frac{3 π}{12})

mixtyggc

Beginner2022-02-05Added 15 answers

Answer:

D = \sqrt{5 - 2 \sqrt{3}} \approx 1.2393

Explanation:
Distance between Polar Co-ordinates:

A (r_{1}, θ_{1}) and B (r_{2}, θ_{2})

D = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2})} \dots \to (I)

We have,

P_{1} (2, \frac{5 π}{12}) and P_{2} (1, \frac{3 π}{12})

So,

r_{1} = 2, r_{2} = 1, θ_{1} = \frac{5 π}{12} and θ_{2} = \frac{3 π}{12}

\Rightarrow θ_{1} - θ_{2} = \frac{5 π}{12} - \frac{3 π}{12} = \frac{2 π}{12} = \frac{π}{6} = 30^{\circ}

\Rightarrow \cos (θ_{1} - θ_{2}) = \cos (30^{\circ})

Using: (I) we get

D = \sqrt{2^{2} + 1^{2} - 2 (2) (1) {\cos 30}^{\circ}}

\Rightarrow D = \sqrt{4 + 1 - 4 \cdot \frac{\sqrt{3}}{2}}

\Rightarrow D = \sqrt{5 - 2 \sqrt{3}}

\Rightarrow D \approx 1.2393

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