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**Unformatted text preview: **e.
Example 11.4.1. For each point in polar coordinates given below plot the point and then give two
additional expressions for the point, one of which has r > 0 and the other with r < 0.
1. P (2, 240◦ ) π
2. P −4, 76 π
3. P 117, − 52 4. P −3, − π
4 Solution.
1. Whether we move 2 units along the polar axis and then rotate 240◦ or rotate 240◦ then move
out 2 units from the pole, we plot P (2, 240◦ ) below.
θ = 240◦
Pole Pole P (2, 240◦ ) We now set about ﬁnding alternate descriptions (r, θ) for the point P . Since P is 2 units from
the pole, r = ±2. Next, we choose angles θ for each of the r values. The given representation
for P is (2, 240◦ ) so the angle θ we choose for the r = 2 case must be coterminal with 240◦ .
(Can you see why?) One such angle is θ = −120◦ so one answer for this case is (2, −120◦ ). For
the case r = −2, we visualize our rotation starting 2 units to the left of the pole. From this
position, we need only to rotate θ = 60◦ to arrive at...

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