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Pre-Calculus

Pre-Calculus. Chapter 6 Additional Topics in Trigonometry. 6.5 Trig Form of a Complex Number. Objectives: Find absolute values of complex numbers. Write trig forms of complex numbers. Multiply and divide complex numbers written in trig form.

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Pre-Calculus

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  1. Pre-Calculus Chapter 6 Additional Topics in Trigonometry

  2. 6.5 Trig Form of a Complex Number Objectives: • Find absolute values of complex numbers. • Write trig forms of complex numbers. • Multiply and divide complex numbers written in trig form. • Use DeMoivre’s Theorem to find powers of complex numbers. • Find nth roots of complex numbers.

  3. Graphical Representation of a Complex Number • Graph in coordinate plane called the complex plane • Horizontalaxisis the realaxis. • Vertical axis is the imaginaryaxis. 3 + 4i• -2 + 3i• • -5i

  4. Absolute Value of a Complex Number • Defined as the length of the line segment from the origin(0, 0) to the point. • Calculate using the Distance Formula. 3 + 4i•

  5. Examples • Graph the complex number. • Find the absolute value.

  6. Trig Form of Complex Number • Graph the complex number. • Notice that a right triangle is formed. a + bi• r b θ a How do we determine θ?

  7. Trig Form of Complex Number • Substitute & into z = a + bi. • Result is • Sometimes abbreviated as

  8. Examples • Write the complex number –5 + 6iin trig form. • r = ? • θ = ? • Write z= 3 cos315° + 3i sin315° in standard form. • r = ? • a = ? • b = ?

  9. Product of Trig Form of Complex Numbers • Given and • It can be shown that the product is • That is, • Multiply the absolute values. • Add the angles.

  10. Quotient of Trig Form of Complex Numbers • Given and • It can be shown that the quotient is • That is, • Divide the absolute values. • Subtract the angles.

  11. Examples • Calculate using trig form and convert answers to standard form.

  12. Powers of Complex Numbers • If z = r (cosθ + i sin θ), find z2. • What about z3?

  13. DeMoivre’s Theorem • If z = r (cosθ + i sin θ) is a complex number and n is a positive integer, then

  14. Examples • Apply DeMoivre’s Theorem.

  15. Roots of Complex Numbers • Recall the Fundamental Theorem of Algebra in which a polynomial equation of degree nhas exactly n complex solutions. • An equation such as x6 = 1 will have six solutions. Each solution is a sixth root of 1. • In general, the complex number u = a + biisan nth root of the complex number z if

  16. Solutions to Previous Example • An equation such as x6 = 1 will have six solutions. Each solution is a sixth root of 1.

  17. nth Roots of a Complex Number • For a positive integer n, the complex number z = r (cosθ + i sin θ) has exactly n distinct nth roots given by • Note: The roots are equally spaced around a circle of radius centered at the origin.

  18. Example • Find the three cube roots of z = –2 + 2i. • Write complex number in trig form. • Find r. • Find θ. • Use the formula with k = 0, 1, and 2.

  19. Solution

  20. Homework 6.5 • Worksheet 6.5

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