Textbook Question
In Exercises 61–63, test for symmetry with respect to
a. the polar axis.
b. the line θ = π/2.
c. the pole.
r = 5 + 3 cos θ
Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5Chapter 5, Problem 5
In Exercises 1–10, plot each complex number and find its absolute value. z = 3 + 2i
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Identify the complex number given: \(z = 3 + 2i\), where the real part is 3 and the imaginary part is 2.
Plot the complex number on the complex plane by marking the point with coordinates \((3, 2)\), where the x-axis represents the real part and the y-axis represents the imaginary part.
Recall that the absolute value (or modulus) of a complex number \(z = a + bi\) is given by the formula \(|z| = \sqrt{a^2 + b^2}\).
Substitute the values of the real part \(a = 3\) and the imaginary part \(b = 2\) into the formula: \(|z| = \sqrt{3^2 + 2^2}\).
Simplify the expression under the square root to find the absolute value, which represents the distance of the point from the origin in the complex plane.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers and the Complex Plane
A complex number is expressed as z = a + bi, where a is the real part and b is the imaginary part. It can be represented as a point or vector in the complex plane, with the horizontal axis for the real part and the vertical axis for the imaginary part.
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Dividing Complex Numbers
Plotting Complex Numbers
To plot a complex number, locate the point corresponding to its real part on the x-axis and its imaginary part on the y-axis. For z = 3 + 2i, plot the point at (3, 2) in the complex plane.
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Absolute Value (Modulus) of a Complex Number
The absolute value or modulus of a complex number z = a + bi is the distance from the origin to the point (a, b) in the complex plane. It is calculated as |z| = √(a² + b²), representing the magnitude of the complex number.
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4:22Dividing Complex Numbers
Related Practice
Textbook Question
Evaluate x²+19 / 2−x for x = 3i.
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Textbook Question
In Exercises 59–62, sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation's domain and range. x = t² + t + 1, y = 2t
Textbook Question
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√3 (cos (5π/18) + i sin (5π/18))]⁶
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Textbook Question
In Exercises 64–70, graph each polar equation. Be sure to test for symmetry. r = 2 + 2 sin θ
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Textbook Question
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Line: Passes through (−2,4) and (1,7)
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