Polar coordinates

Three quick questions from earlier lessons.

A lighthouse beam points at $90^\circ$ (straight north) and a boat is 3 km away along it. In rectangular coordinates, where is the boat?

Polar coordinates $(r, \theta)$ give a point by its distance $r$ from the origin and its angle $\theta$ from the positive $x$ -axis.

x = r\cos\theta

,

y = r\sin\theta

. Back:

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

and

\tan\theta = \dfrac{y}{x}

, with

\theta

in the quadrant of the point.

Convert $(\sqrt{3}, -1)$ to polar coordinates.

r = \sqrt{3 + 1} = 2

Distance from the origin.

\theta = 2\pi - \dfrac{\pi}{6} = \dfrac{11\pi}{6}

tan θ = −1/√3; the point is right and down: quadrant IV.

\left(2, \dfrac{11\pi}{6}\right)

Both coordinates.

Your turn: convert $\left(4, \dfrac{\pi}{3}\right)$ to rectangular.

x = 2

4 cos(π/3) = 4 · 1/2.

y = 2\sqrt{3}

4 sin(π/3) = 4 · √3/2.

Convert the polar point $\left(7, \frac{\pi}{6}\right)$ to rectangular coordinates (exact).
Convert the polar point $\left(8, \frac{5 \pi}{3}\right)$ to rectangular coordinates (exact).
Convert the polar point $\left(7, \frac{2 \pi}{3}\right)$ to rectangular coordinates (exact).

Convert the point $(-2, 2)$ to polar coordinates $(r, \theta)$ with $r > 0$ and $0 \le \theta < 2\pi$ .
Convert the point $(-5, 5)$ to polar coordinates $(r, \theta)$ with $r > 0$ and $0 \le \theta < 2\pi$ .

Three questions to check you have got it.

Convert the polar point $\left(7, \frac{\pi}{6}\right)$ to rectangular coordinates (exact).
Convert the polar point $\left(8, \frac{5 \pi}{3}\right)$ to rectangular coordinates (exact).
Convert the polar point $\left(7, \frac{2 \pi}{3}\right)$ to rectangular coordinates (exact).

Polar coordinates

Watch out: choose $\theta$ in the quadrant of the point.

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