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

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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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