Presentation on theme: "Pre-Calculus Chapter 6 Additional Topics in Trigonometry."— Presentation transcript:
1Pre-CalculusChapter 6Additional Topics in Trigonometry
26.5 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.
3Graphical 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
4Absolute 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 •
5ExamplesGraph the complex number.Find the absolute value.
6Trig Form of Complex Number Graph the complex number.Notice that a right triangle is formed.θa + bi •barHow do we determine θ?
7Trig Form of Complex Number Substitute & into z = a + bi.Result isSometimes abbreviated as
8Examples Write the complex number –5 + 6i in trig form. r = ? θ = ? Write z = 3 cos 315° + 3i sin 315° in standard form.a = ?b = ?
9Product of Trig Form of Complex Numbers GivenandIt can be shown that the product isThat is,Multiply the absolute values.Add the angles.
10Quotient of Trig Form of Complex Numbers GivenandIt can be shown that the quotient isThat is,Divide the absolute values.Subtract the angles.
11ExamplesCalculate using trig form and convert answers to standard form.
12Powers of Complex Numbers If z = r (cos θ + i sin θ), find z2.What about z3?
13DeMoivre’s TheoremIf z = r (cos θ + i sin θ) is a complex number and n is a positive integer, then
15Roots 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
16Solutions to Previous Example An equation such as x6 = 1 will have six solutions. Each solution is a sixth root of 1.
17nth 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.
18Example 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.