The equivalent polar coordinates for the given rectangular coordinates: \[( 5,\ 180^{\circ} )\]

Question

The equivalent polar coordinates for the given rectangular coordinates:

[\begin{array}{cc} 5 & 180^{\circ} \end{array}]

Answer 1

Concepts used: Conversion formula for coordinate systems are given as:
a) From polar to rectangular:

[x = r \cos θ y = r \sin θ]

b) From rectangular to polar:

r = \pm \sqrt{x^{2} + y^{2}}

\cos θ = \frac{x}{r}

\sin θ = \frac{y}{r}

Calculation:
Given rectangular coordinate is

[\begin{array}{cc} 5 & 180^{\circ} \end{array}]

.
Here

[r = 5, θ = 180^{\circ}]

.
Converrting into equivalent rectangular coordinates:

x = r \cos θ

\Rightarrow x = 5 \cos 180^{\circ}

\Rightarrow x = 5 (- 1)

\Rightarrow x = - 5

And

[y = r \sin θ

\Rightarrow y = 5 \sin 180^{\circ}

\Rightarrow y = 5 (0)

\Rightarrow y = 0

Hence, desired equivalent rectangular coordinates would be

[\begin{array}{cc} - 5 & 0 \end{array}] .

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