Presentation on theme: "Pre-Calculus Chapter 6 Additional Topics in Trigonometry."— Presentation transcript:
1 Pre-CalculusChapter 6Additional 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 planeHorizontal axis is the real axis.Vertical axis is the imaginary axis.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 ExamplesGraph the complex number.Find the absolute value.
6 Trig Form of Complex Number Graph the complex number.Notice that a right triangle is formed.θa + bi •barHow do we determine θ?
7 Trig Form of Complex Number Substitute & into z = a + bi.Result isSometimes abbreviated as
8 Examples Write the complex number –5 + 6i in trig form. r = ? θ = ? Write z = 3 cos 315° + 3i sin 315° in standard form.a = ?b = ?
9 Product of Trig Form of Complex Numbers GivenandIt can be shown that the product isThat is,Multiply the absolute values.Add the angles.
10 Quotient of Trig Form of Complex Numbers GivenandIt can be shown that the quotient isThat is,Divide the absolute values.Subtract the angles.
11 ExamplesCalculate using trig form and convert answers to standard form.
12 Powers of Complex Numbers If z = r (cos θ + i sin θ), find z2.What about z3?
13 DeMoivre’s TheoremIf z = r (cos θ + i sin θ) is a complex number and n is a positive integer, then
15 Roots of Complex Numbers Recall the Fundamental Theorem of Algebra in which a polynomial equation of degree n has 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 + bi is an 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 (cos θ + i sin θ) has exactly n distinct nth roots given byNote: 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.