Complex Numbers and DeMoivre’s Theorem LESSON 9–5 Complex Numbers and DeMoivre’s Theorem

Five-Minute Check (over Lesson 9-4) Then/Now New Vocabulary Key Concept: Absolute Value of a Complex Number Example 1: Graphs and Absolute Values of Complex Numbers Key Concept: Polar Form of a Complex Number Example 2: Complex Numbers in Polar Form Example 3: Graph and Convert the Polar Form of a Complex Number Key Concept: Product and Quotient of Complex Numbers in Polar Form Example 4: Product of Complex Numbers in Polar Form Example 5: Real-World Example: Quotient of Complex Numbers in Polar Form Key Concept: De Moivre’s Theorem Example 6: De Moivre’s Theorem Key Concept: Distinct Roots Example 7: pth Roots of a Complex Number Example 8: The pth Roots of Unity Lesson Menu

Determine the eccentricity, type of conic, and equation of the directrix for the polar equation . A. e = 4; ellipse; x = 3 B. e = 2; ellipse; y = –3 C. e = 4; hyperbola; x = 12 D. e = 2; hyperbola; x = –3 5–Minute Check 1

Determine the eccentricity, type of conic, and equation of the directrix for the polar equation . A. e = 4; hyperbola; y = 8 B. e = 4; ellipse; x = 2 C. e = 1; parabola; y = 2 D. e = 1; parabola; y = –2 5–Minute Check 2

A. Write a polar equation and directrix for the conic with e = 1 and directrix y = −2. B. C. D. 5–Minute Check 3

B. Graph a polar equation and directrix for the conic with e = 1 and directrix y = −2. 5–Minute Check 3

Write in rectangular form. B. C. r + x – 3 = 0 D. x 2 = 9 – 6y 5–Minute Check 4

What type of conic is given by A. parabola B. ellipse C. hyperbola D. circle 5–Minute Check 5

Convert complex numbers from rectangular to polar form and vice versa. 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

absolute value of a complex number polar form trigonometric form 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

Key Concept 1

A. Graph z = 2 + 3i in the complex plane and find its absolute value. 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

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

B. Graph z = –3 + i in the complex plane and find its absolute value. 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

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

Graph 3 – 4i in the complex plane and find its absolute value. Example 1

Key Concept 2

A. Express the complex number –2 + 5i in polar form. Complex Numbers in Polar Form A. Express the complex number –2 + 5i in polar form. Find the modulus r and argument . Conversion formula a = –2 and b = 5 Simplify. 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

B. Express the complex number 6 + 2i in polar form. Complex Numbers in Polar Form B. Express the complex number 6 + 2i in polar form. Find the modulus r and argument . Conversion formula a = 6 and b = 2 Simplify. 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

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

Graph on a polar grid. Then express it in rectangular form. Graph and Convert the Polar Form of a Complex Number 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 Example 3

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

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

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

Key Concept 4

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

Now find the rectangular form of the product. 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 + 0i or –10. Answer: 10(cos π + i sin π); –10 Example 4

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

Express each number in polar form. 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. 100 = 100(cos 0 +j sin 0) 4 – 3j = 5[cos (–0.64) + jsin (–0.64)] Example 5

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

Now, convert the current to rectangular form. Quotient of Complex Numbers in Polar Form Quotient Formula Simplify. 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

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

Key Concept 6

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

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

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

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

Key Concept 7

Find the fifth roots of –2 – 2i. pth Roots of a Complex Number Find the fifth roots of –2 – 2i. First, write –2 – 2i in polar form. –2 – 2i = Now, write an expression for the fifth roots. Example 7

Let n = 0, 1, 2, 3, and 4 successively to find the fifth roots. pth Roots of a Complex Number Simplify. Let n = 0, 1, 2, 3, and 4 successively to find the fifth roots. Example 7

Let n = 0 First fifth root Let n = 1 pth Roots of a Complex Number Example 7

Second fifth root Let n = 2 Third fifth root pth Roots of a Complex Number Second fifth root Let n = 2 Third fifth root Example 7

Let n = 3 Fourth fifth root Let n = 4 Fifth fifth root pth Roots of a Complex Number Let n = 3 Fourth fifth root Let n = 4 Fifth fifth root Example 7

pth Roots of a Complex Number The fifth roots of –2 – 2i are approximately 0.87 + 0.87i, –0.56 + 1.10i, –1.22 – 0.19i, –0.19 – 1.22i, and 1.10 – 0.56i. Answer: 0.87 + 0.87i, –0.56 + 1.10i, –1.22 – 0.19i, –0.19 – 1.22i, 1.10 – 0.56i Fifth fifth root Example 7

Find the cube roots of A. –1.04 + 5.91i, –4.60 – 3.86i, 5.64 – 2.05i B. 3 – 5.20i, 3 + 5.20i, 5.64  2.05i C. –0.32 + 1.79i, –1.39 – 1.17i, 1.71 – 0.62i D. 0.91 – 1.57i, 4.60 + 3.86i , –1.82 Example 7

Find the fifth roots of unity. The pth Roots of Unity Find the fifth roots of unity. First write 1 in polar form. 1 = 1 • (cos 0 + i sin 0) Now write an expression for the fifth roots. Simplify. Example 8

Let n = 0 to find the first root of 1. The pth Roots of Unity Let n = 0 to find the first root of 1. Let n = 0 First Roots Notice that the modulus of each complex number is 1. The arguments are found by , resulting in  increasing by for each successive root. Therefore, we can calculate the remaining roots by adding to each previous . Example 8

cos 0 + i sin 0 or 1 1st root 2nd root 3rd root 4th root 5th root The pth Roots of Unity cos 0 + i sin 0 or 1 1st root 2nd root 3rd root 4th root 5th root Example 8

The pth Roots of Unity The fifth roots of 1 are approximately 1, 0.31 + 0.95i, –0.81 + 0.59i, –0.81 – 0.59i, and 0.31 – 0.95i, as shown in the figure. Answer: 1, 0.31 + 0.95i, –0.81 + 0.59i, –0.81 – 0.59i, 0.31 – 0.95i Example 8

Find the fourth roots of unity. A. B. 0.92 + 0.38i, –0.38 + 0.92i, –0.92 – 0.38i, 0.38 – 0.92i C. D. 1, i, –1, –i Example 8

Complex Numbers and DeMoivre’s Theorem LESSON 9–5 Complex Numbers and DeMoivre’s Theorem