Given and , find the quotient .
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
11. Graphing Complex Numbers
Products and Quotients of Complex Numbers
Problem 19
Textbook Question
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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Complex Numbers In Polar Form
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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Introduction to Trigonometric Functions
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