Given and , find the product .
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 5.2.45
Textbook Question
In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.
z₁ = 20(cos 75° + i sin 75°)
z₂ = 4(cos 25° + i sin 25°)
Verified step by step guidance1
Recall that when dividing two complex numbers in polar form, \(z_1 = r_1(\cos \theta_1 + i \sin \theta_1)\) and \(z_2 = r_2(\cos \theta_2 + i \sin \theta_2)\), the quotient is given by:
\(\frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right)\)
Identify the magnitudes and arguments of the given complex numbers:
\(r_1 = 20\), \(\theta_1 = 75^\circ\)
\(r_2 = 4\), \(\theta_2 = 25^\circ\)
Calculate the magnitude of the quotient by dividing the magnitudes:
\(r = \frac{r_1}{r_2} = \frac{20}{4}\)
Calculate the argument of the quotient by subtracting the angles:
\(\theta = \theta_1 - \theta_2 = 75^\circ - 25^\circ\)
Write the quotient in polar form using the results from steps 3 and 4:
\(\frac{z_1}{z_2} = r \left( \cos \theta + i \sin \theta \right)\), where \(r\) and \(\theta\) are the values found above. Make sure the argument \(\theta\) is expressed between \(0^\circ\) and \(360^\circ\) if necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Numbers in Polar Form
Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). This form simplifies multiplication and division by working directly with magnitudes and angles instead of real and imaginary parts.
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Complex Numbers In Polar Form
Division of Complex Numbers in Polar Form
To divide two complex numbers in polar form, divide their magnitudes and subtract the arguments: (r₁/r₂)(cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)). This method avoids converting back to rectangular form and makes calculations more straightforward.
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Complex Numbers In Polar Form
Argument of a Complex Number and Angle Normalization
The argument of a complex number is the angle it makes with the positive real axis. When expressing the argument, it is often normalized to lie within 0° to 360° by adding or subtracting full rotations (360°) to ensure a standard, positive angle measurement.
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Dividing Complex Numbers
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