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Section 7.4 Polar coordinates (CO4)

Subsection 7.4.1 Activities

A point in the polar coordinate system.
Figure 166. A point in the polar coordinate system.
The polar grid
Figure 167. The polar grid.

Activity 7.4.2.

(a)
Plot the Cartesian point \(P=(x,y)=(\sqrt{3},-1)\) and draw line segments connecting the origin to \(P\text{,}\) the origin to \((x,y)=(\sqrt{3},0)\text{,}\) and \(P\) to \((x,y)=(\sqrt{3},0)\text{.}\)
(b)
Solve the triangle formed by the line segments you just drew (i.e. find the lengths of all sides and the measures of each angle).
(c)
Find all polar coordinates for the Cartesian point \((x,y)=(\sqrt{3},-1)\text{.}\)
(d)
Find Cartesian coordinates for the polar point \((r,\theta)=\left(-\sqrt{2},\displaystyle\frac{3\pi}{4}\right)\text{.}\)

Activity 7.4.3.

Graph each of the following.
(a)
\(r=1\)
(b)
\(r=-1\)
(c)
\(\theta=\displaystyle\frac{\pi}{6}\)
(d)
\(\theta=\displaystyle\frac{7\pi}{6}\)
(e)
\(\theta=\displaystyle\frac{-5\pi}{6}\)
(f)
\(1\leq r < -1\text{,}\) \(0\leq\theta\leq\displaystyle\frac{\pi}{2}\)
(g)
\(-3\leq r \leq 2\text{,}\) \(\theta=\displaystyle\frac{\pi}{4}\)
(h)
\(r \leq 0\text{,}\) \(\theta=\displaystyle\frac{-\pi}{2}\)
(i)
\(\displaystyle\frac{2\pi}{3}\leq\theta\leq\displaystyle\frac{5\pi}{6}\)
(j)
\(r=3\sec(\theta)\)

Activity 7.4.7.

(a)
Find a polar form of the the Cartesian equation \(x^2+(y-3)^2=9\text{.}\)

Activity 7.4.8.

Find a Cartesian form of each of the given polar equations.
(a)
\(r^2=4r\cos(\theta)\)
(b)
\(r=\displaystyle\frac{4}{2\cos(\theta)-\sin(\theta)}\)