Textbook Question
In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−2, − π/2)
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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 19
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 19Chapter 5, Problem 19
In Exercises 19–21, find the product of the complex numbers. Leave answers in polar form.
z₁ = 3(cos 40°+i sin 40°)
z₂ = 5(cos 70°+i sin 70°)
Verified step by step guidance1
Recall that when multiplying two complex numbers in polar form, the magnitudes multiply and the angles add. Specifically, if \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \), then their product is \( z_1 z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right) \).
Identify the magnitudes and angles from the given complex numbers: \( r_1 = 3 \), \( \theta_1 = 40^\circ \), \( r_2 = 5 \), and \( \theta_2 = 70^\circ \).
Multiply the magnitudes: calculate \( r = r_1 \times r_2 = 3 \times 5 \).
Add the angles: calculate \( \theta = \theta_1 + \theta_2 = 40^\circ + 70^\circ \).
Write the product in polar form using the results from the previous steps: \( z_1 z_2 = r \left( \cos \theta + i \sin \theta \right) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Form of Complex Numbers
A complex number can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude (modulus) and θ is the argument (angle). This form is useful for multiplication and division because it separates the magnitude and angle components.
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Complex Numbers In Polar Form
Multiplication of Complex Numbers in Polar Form
When multiplying two complex numbers in polar form, multiply their magnitudes and add their angles. Specifically, if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁z₂ = r₁r₂ [cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)].
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Trigonometric Functions and Angle Addition
Understanding cosine and sine functions and how angles add is essential. The angle addition in the product uses the sum of the individual arguments, which relies on the properties of trigonometric functions to combine the angles correctly.
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Related Practice
Textbook Question
In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −3
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Textbook Question
In Exercises 9–20, use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. x = 2t, y = |t − 1|; −∞ < t < ∞
Textbook Question
In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 + 2 cos θ
Textbook Question
In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 + cos θ
Textbook Question
In Exercises 9–20, find each product and write the result in standard form. (2 + 3i)²