Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
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- In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (5, π/6)
Problem 21
- In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 + 2 cos θ
Problem 21
Problem 22
In Exercises 21–28, divide and express the result in standard form.
3 / 4+i
- In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 − 3 sin θ
Problem 23
- In Exercises 21–28, divide and express the result in standard form. 8i / 4−3i
Problem 25
- In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. _ x = √t, y = t − 1
Problem 25
- In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (4, π/2)
Problem 25
- In Exercises 27–32, select the representations that do not change the location of the given point. (7, 140°) (−7, 320°)
Problem 27
- In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ
Problem 27
- In Exercises 27–32, select the representations that do not change the location of the given point. (4, 120°) (−4, 300°)
Problem 28
- In Exercises 29–36, simplify and write the result in standard form. ___ √−49
Problem 29
- In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = 1 + 3 cos t, y = 2 + 3 sin t; 0 ≤ t < 2π
Problem 29
- In Exercises 27–32, select the representations that do not change the location of the given point. (2, − 3π/4) (2, − 7π/4)
Problem 29
- In Exercises 27–32, select the representations that do not change the location of the given point. (−2, 7π/6) (−2, −5π/6)
Problem 30
- In Exercises 29–36, simplify and write the result in standard form. ____ √−108
Problem 31
- In Exercises 27–32, select the representations that do not change the location of the given point. (−5, − π/4) (−5, 7π/4)
Problem 31
- In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − 3 sin θ
Problem 31
- In Exercises 27–32, select the representations that do not change the location of the given point. (−6, 3π) (6, −π)
Problem 32
- In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (4, 90°)
Problem 33
- In Exercises 13–34, test for symmetry and then graph each polar equation. r cos θ = −3
Problem 33
Problem 34
In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.
x = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π
- In Exercises 29–36, simplify and write the result in standard form. ____________ √1² − 4 ⋅ 0.5 ⋅ 5
Problem 35
- In Exercises 35–44, test for symmetry and then graph each polar equation. r = cos θ/2
Problem 35
Problem 36
In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.
x = 5 sec t, y = 3 tan t
- In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ ___ √−64 − √−25
Problem 37
- In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (−4, π/2)
Problem 37
- In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ ___ 5√−16 + 3√−81
Problem 39
- In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = 2ᵗ, y = 2⁻ᵗ; t ≥ 0
Problem 39
- In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (7.4, 2.5)
Problem 39
- In Exercises 35–44, test for symmetry and then graph each polar equation. r = 1 / 1−cos θ
Problem 39