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Complex Numbers: Rectangular & Polar Forms

Learn how to convert complex numbers between rectangular and polar forms, find products, quotients, powers, and roots. Explore absolute values, polar and trigonometric forms, and more concepts.

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Complex Numbers: Rectangular & Polar Forms

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  1. Splash Screen

  2. Find rectangular coordinates for the point with the given polar coordinates. A. B. C. D.(0, 3) 5–Minute Check 1

  3. Find rectangular coordinates for the point with the given polar coordinates. A. B. C. D.(0, 3) 5–Minute Check 1

  4. You performed operations with complex numbers written in rectangular form. (Lesson 0-6) • Convert complex numbers from rectangular to polar form and vice versa. • Find products, quotients, powers, and roots of complex numbers in polar form. Then/Now

  5. complex plane • real axis • imaginary axis • Argand plane • absolute value of a complex number • polar form • trigonometric form • modulus • argument • pth roots of unity Vocabulary

  6. Key Concept 1

  7. Graphs and Absolute Values of Complex Numbers A. Graph z = 2 + 3i in the complex plane and find its absolute value. (a, b) = (2, 3) Example 1

  8. Absolute value formula a = 2 and b = 3 Simplify. The absolute value of 2 + 3i is Graphs and Absolute Values of Complex Numbers Answer: Example 1

  9. Absolute value formula a = 2 and b = 3 Simplify. Answer: The absolute value of 2 + 3i is Graphs and Absolute Values of Complex Numbers Example 1

  10. Graphs and Absolute Values of Complex Numbers B. Graph z = –3 + i in the complex plane and find its absolute value. (a, b) = (–3, 1) Example 1

  11. Absolute value formula a = –3 and b = 1 Simplify. The absolute value of –3 + i is Graphs and Absolute Values of Complex Numbers Answer: Example 1

  12. Absolute value formula a = –3 and b = 1 Simplify. The absolute value of –3 + i is Answer: Graphs and Absolute Values of Complex Numbers Example 1

  13. A. 5; B. 5; C. 1; D. 7; Graph 3 – 4i in the complex plane and find its absolute value. Example 1

  14. A. 5; B. 5; C. 1; D. 7; Graph 3 – 4i in the complex plane and find its absolute value. Example 1

  15. Key Concept 2

  16. Conversion formula a = –2 and b = 5 Simplify. Complex Numbers in Polar Form A. Express the complex number –2 + 5i in polar form. Find the modulus r and argument . The polar form of –2 + 5i is about 5.39(cos 1.95 + i sin 1.95). Answer: Example 2

  17. Conversion formula a = –2 and b = 5 Simplify. Complex Numbers in Polar Form A. Express the complex number –2 + 5i in polar form. Find the modulus r and argument . The polar form of –2 + 5i is about 5.39(cos 1.95 + i sin 1.95). Answer:5.39(cos 1.95 + i sin 1.95) Example 2

  18. Conversion formula a = 6 and b = 2 Simplify. Complex Numbers in Polar Form B. Express the complex number 6 + 2i in polar form. Find the modulus r and argument . The polar form of 6 + 2i is about 6.32(cos 0.32 +i sin 0.32). Answer: Example 2

  19. Conversion formula a = 6 and b = 2 Simplify. Complex Numbers in Polar Form B. Express the complex number 6 + 2i in polar form. Find the modulus r and argument . The polar form of 6 + 2i is about 6.32(cos 0.32 +i sin 0.32). Answer:6.32(cos 0.32 + i sin 0.32) Example 2

  20. Express the complex number 4 – 5i in polar form. A. 20(cos 5.61 + i sin 5.61) B. 20(cos 0.90 + i sin 0.90) C. 6.40(cos 4.04 + i sin 4.04) D. 6.40(cos 5.39 + i sin 5.39) Example 2

  21. Express the complex number 4 – 5i in polar form. A. 20(cos 5.61 + i sin 5.61) B. 20(cos 0.90 + i sin 0.90) C. 6.40(cos 4.04 + i sin 4.04) D. 6.40(cos 5.39 + i sin 5.39) Example 2

  22. Graph on a polar grid. Then express it in rectangular form. The value of r is 4, and the value of  is Plot the polar coordinates Graph and Convert the Polar Form of a Complex Number Example 3

  23. Polar form Evaluate for cosine and sine. Distributive Property The rectangular form of Graph and Convert the Polar Form of a Complex Number To express the number in rectangular form, evaluate the trigonometric values and simplify. Example 3

  24. Graph and Convert the Polar Form of a Complex Number Answer: Example 3

  25. Graph and Convert the Polar Form of a Complex Number Answer: Example 3

  26. Express in rectangular form. A. –6 – 6i B. C. D. Example 3

  27. Express in rectangular form. A. –6 – 6i B. C. D. Example 3

  28. Key Concept 4

  29. Find in polar form. Then express the product in rectangular form. Original expression Product Formula Simplify. Product of Complex Numbers in Polar Form Example 4

  30. Product of Complex Numbers in Polar Form Now find the rectangular form of the product. 10(cos π + i sin π) Polar form = 10(–1 + 0i) Evaluate. = –10 + 0i Distributive Property The polar form of the product is 10(cos π + i sin π). The rectangular form of the product is –10 + 0ior –10. Answer: Example 4

  31. Product of Complex Numbers in Polar Form Now find the rectangular form of the product. 10(cos π + i sin π) Polar form = 10(–1 + 0i) Evaluate. = –10 + 0i Distributive Property The polar form of the product is 10(cos π + i sin π). The rectangular form of the product is –10 + 0ior –10. Answer:10(cos π + i sin π); –10 Example 4

  32. Find Express your answer in rectangular form. A. –7.25 + 27.05i B. –19.80 – 19.80i C. –27.05 + 7.25i D. –10.63 + 2.85i Example 4

  33. Find Express your answer in rectangular form. A. –7.25 + 27.05i B. –19.80 – 19.80i C. –27.05 + 7.25i D. –10.63 + 2.85i Example 4

  34. 100 = 100(cos 0 +j sin 0) 4 – 3j = 5[cos (–0.64) + jsin (–0.64)] Quotient of Complex Numbers in Polar Form ELECTRICITY If a circuit has a voltage E of 100 volts and an impedance Z of 4 – 3j ohms, find the current I in the circuit in rectangular form. Use E = I •Z. Express each number in polar form. Example 5

  35. Divide each side by Z. E = 100(cos 0 + j sin 0) Z = 5[cos (–0.64) + j sin (–0.64)] Quotient of Complex Numbers in Polar Form Solve for the current I in E = I•Z. I•Z = E Original equation Example 5

  36. Quotient Formula Simplify. Quotient of Complex Numbers in Polar Form Now, convert the current to rectangular form. I = 20(cos 0.64 + j sin 0.64) Original equation = 20(0.80 + 0.60j) Evaluate. = 16.04 + 11.94j Distributive Property The current is about 16.04 + 11.94j amps. Answer: Example 5

  37. Quotient Formula Simplify. Quotient of Complex Numbers in Polar Form Now, convert the current to rectangular form. I = 20(cos 0.64 + j sin 0.64) Original equation = 20(0.80 + 0.60j) Evaluate. = 16.04 + 11.94j Distributive Property The current is about 16.04 + 11.94j amps. Answer:16.04 + 11.94j amps Example 5

  38. ELECTRICITY If a circuit has a voltage of 140 volts and a current of 4 + 3j amps, find the impedance of the circuit in rectangular form. A. 0.03 + 0.02j B. 22.4 – 16.8j C. 560 + 420j D. 23.4 + 16.87j Example 5

  39. ELECTRICITY If a circuit has a voltage of 140 volts and a current of 4 + 3j amps, find the impedance of the circuit in rectangular form. A. 0.03 + 0.02j B. 22.4 – 16.8j C. 560 + 420j D. 23.4 + 16.87j Example 5

  40. Key Concept 6

  41. Find and express in rectangular form. First, write in polar form. Conversion formula a = 3 and b = Simplify. Simplify. De Moivre’s Theorem Example 6

  42. The polar form of is Original equation De Moivre's Theorem Simplify. De Moivre’s Theorem Now use De Moivre's Theorem to find the fourth power. Example 6

  43. Evaluate. Simplify. Therefore, De Moivre’s Theorem Answer: Example 6

  44. Evaluate. Simplify. Therefore, Answer: De Moivre’s Theorem Example 6

  45. Find and express in rectangular form. A. 1728i B. 1728 C. D. Example 6

  46. Find and express in rectangular form. A. 1728i B. 1728 C. D. Example 6

  47. complex plane • real axis • imaginary axis • Argand plane • absolute value of a complex number • polar form • trigonometric form • modulus • argument • pth roots of unity Vocabulary

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