9.4 POLAR COORDINATES
Rectangular coordinates
Polar coordinates
Exercise 4.1 Plotting Points in Polar Coordinates and finding coordinates (x, y).
Plot the points with the indicated polar coordinates (r, θ) and determine the
corresponding rectangular coordinates (x, y) for:
(a) (2, 0)
(b)
(c)
MTH 2103 – NOTES & EXERCISES 9.4
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Example 4.6 Converting an Equation from Rectangular to Polar Coordinates
Find the polar equation(s) corresponding to the hyperbola x2 − y2 = 9.
Solution:
𝑥2 − 𝑦2 = 9
𝑏𝑢𝑡
𝑥 = 𝑟 𝑐𝑜𝑠𝜃 𝑎𝑛𝑑
𝑦 = 𝑟 𝑠𝑖𝑛𝜃
(𝑟 cos 𝜃)2 − (𝑟 sin 𝜃 )2 = 9
𝑟 2 cos 2 𝜃 − 𝑟 2 sin2 𝜃 = 9
𝑟 2 (cos 2 𝜃 − sin2 𝜃) = 9
𝑟2 cos 2𝜃 = 9
9
𝑟 =
cos 2𝜃
2
𝑟 2 = 9 sec 2𝜃
𝑟 = ±3√sec 2𝜃
 polar equation
MTH 2103 – NOTES & EXERCISES 9.4
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In exercises 1–16, find their rectangular coordinates for the given polar
point
3. ( − 2, π)
13.
16.
4.
𝜋
(2, − 3 )
𝜋
(3, 8 )
MTH 2103 – NOTES & EXERCISES 9.4
(−3,
14.
3𝜋
2
)
𝜋
(−1, 3 )
18. (− 3, 1)
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In exercises 51–56, find a polar equation corresponding to the given
rectangular equation.
51.
𝑦2 − 𝑥2 = 4
52.
53.
𝑥 2 + 𝑦 2 = 16𝑦
54.
55.
𝑦=3
MTH 2103 – NOTES & EXERCISES 9.4
56.
𝑥 2 + 𝑦 2 = −9𝑥
𝑥2 + 𝑦2 = 𝑥 + 2𝑦
𝑥=2
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