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Complex numbers precalculus
This document provides information about a lecture on additional topics in trigonometry. It includes definitions of polar coordinates and the relationship between polar and rectangular coordinates. It also discusses representing complex numbers in polar form using De Moivre's theorem and finding nth roots of complex numbers. Key topics covered are plotting points in polar coordinates, converting between polar and rectangular coordinates, graphing complex numbers in the complex plane, writing complex numbers in polar form, multiplying and dividing complex numbers using De Moivre's theorem, and finding nth roots of a complex number.
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Introduction to Chapter 6 on Additional Topics in Trigonometry with information on class schedule.
Explains the concept of Polar Coordinates, including distance and angle definitions.
Examples on plotting points in polar coordinates and explanation of the negative radius concept.
Discussion on the relationship between polar and rectangular coordinates and caution on their usage.
Introduction to graphing complex numbers on the complex plane and calculating their absolute values.
Definition and examples for finding the modulus of complex numbers.
Introduction to the polar form of complex numbers and its relation to sine and cosine.
Examples on converting complex numbers into polar form including calculations of the modulus and argument.
Overview of operations on complex numbers in polar form using De Moivre's Theorem with examples.
Explanation of De Moivre's Theorem for raising complex numbers to powers with examples.
Discussion on the nth roots of complex numbers using De Moivre's Theorem and examples of finding roots.
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Complex numbers precalculus
- 1. CHAPTER 6 Additional Topicsin Trigonometry 1
- 2. 2 INFORMATION – Wednesday26th July 2017 •Lecture on Thursday 27 July on par. 6.5 (continue). •No Lecture on Wednesday 9 August due to the Public Holiday. •Lecture on Thursday 10 August on par. 10.2. •Class Test 1 scheduled for Thursday 3 August on par. 6.5. •No Tutorial on Thursday 5th October (Follows a Monday Timetable)
- 3. 6.5 Trigonometric Form of aComplex Number 4
- 4. 4 Objectives ►Definition of PolarCoordinates ►Relationship Between Polar and Rectangular Coordinates ►Graphing Complex Numbers ►Polar Form of Complex Numbers ►De Moivre’s Theorem ►nth Roots of Complex Numbers
- 5. 5 Definition of PolarCoordinates
- 6. 6 Definition of PolarCoordinates The polar coordinate system uses distances and directions to specify the location of a point in the plane. To set up this system, we choose a fixed point O in the plane called the pole (or origin) and draw from O a ray (half-line) called the polar axis as in the figure.
- 7. 7 Definition of PolarCoordinates Then each point P can be assigned polar coordinates P(r, ) where r is the distance from O to P is the angle between the polar axis and the segment We use the convention that is positive if measured in a counterclockwise direction from the polar axis or negative if measured in a clockwise direction.
- 8. 8 Definition of PolarCoordinates If r is negative, then P(r, ) is defined to be the point that lies |r| units from the pole in the direction opposite to that given by .
- 9. 9 Example 1 –Plotting Points in Polar Coordinates Plot the points whose polar coordinates are given. (a) (1, 3/4) (b) (3, –/6) (c) (3, 3) (d) ( –4, /4) Solution: The points are plotted. (a) (b) (c) (d)
- 10. 10 Example 1 –Solution Note that the point in part (d) lies 4 units from the origin along the angle 5 /4, because the given value of r is negative. cont’d
- 11. 11 Relationship Between Polar andRectangular Coordinates
- 12. 12 Relationship Between Polarand Rectangular Coordinates The connection between the two systems is illustrated in the figure, where the polar axis coincides with the positive x-axis.
- 13. 13 Relationship Between Polarand Rectangular Coordinates The formulas are obtained from the figure using the definitions of the trigonometric functions and the Pythagorean Theorem.
- 14. 14 Example 2 –Converting Polar Coordinates to Rectangular Coordinates
- 15. 15 Relationship Between Polarand Rectangular Coordinates Note that the equations relating polar and rectangular coordinates do not uniquely determine r or . When we use these equations to find the polar coordinates of a point, we must be careful that the values we choose for r and give us a point in the correct quadrant.
- 16.
- 17. 17 Graphing Complex Numbers Tograph real numbers or sets of real numbers, we have been using the number line, which has just one dimension. Complex numbers, however, have two components: a real part and an imaginary part. This suggests that we need two axes to graph complex numbers: one for the real part and one for the imaginary part. We call these the real axis and the imaginary axis, respectively. The plane determined by these two axes is called the complex plane.
- 18. 18 Graphing Complex Numbers Tograph the complex number a + bi, we plot the ordered pair of numbers (a, b) in this plane, as indicated.
- 19. 19 Example 3 –Graphing Complex Numbers Graph the complex numbers z1 = 2 + 3i, z2 = 3 – 2i, and z1 + z2. Solution: We have z1 + z2 = (2 + 3i) + (3 – 2i) = 5 + i. The graph is shown.
- 20. 20 Graphing Complex Numbers Wedefine absolute value for complex numbers in a similar fashion. Using the Pythagorean Theorem, we can see from the figure that the distance between a + bi and the origin in the complex plane is .
- 21. 21 Graphing Complex Numbers Thisleads to the following definition.
- 22. 22 Example 4 –Calculating the Modulus Find the moduli of the complex numbers 3 + 4i and 8 – 5i. Solution: |3 + 4i| = = = 5 |8 – 5i| = =
- 23. 23 Polar Form ofComplex Numbers
- 24. 24 Polar Form ofComplex Numbers Let z = a + bi be a complex number, and in the complex plane let’s draw the line segment joining the origin to the point a + bi . The length of this line segment is r = | z | = .
- 25. 25 Polar Form ofComplex Numbers If is an angle in standard position whose terminal side coincides with this line segment, then by the definitions of sine and cosine a = r cos and b = r sin so z = r cos + ir sin = r (cos + i sin )
- 26. 26 Polar Form ofComplex Numbers We have shown the following.
- 27. 27 Example 5 –Writing Complex Numbers in Polar Form Write each complex number in polar form. (a) 1 + i (b) –1 + (c) –4 – 4i (d) 3 + 4i
- 28. 28 Example 5 –Solution These complex numbers are graphed, which helps us find their arguments. (a) (b) (c) (d)
- 29. 29 Example 5(a) –Solution An argument is = /4 and r = = . Thus
- 30. 30 Example 5(b) –Solution An argument is = 2/3 and r = = 2. Thus cont’d
- 31. 31 Example 5(c) –Solution An argument is = 7/6 (or we could use = –5/6), and r = = 8. Thus cont’d
- 32. 32 Example 5(d) –Solution An argument is = tan–1 and r = = 5. So 3 + 4i = 5 cont’d
- 33. 33 Polar Form ofComplex Numbers This theorem says: To multiply two complex numbers, multiply the moduli and add the arguments To divide two complex numbers, divide the moduli and subtract the arguments.
- 34. 34 Example 6 –Multiplying and Dividing Complex Numbers Let and Find (a) z1z2 and (b) z1/z2. Solution: (a) By the Multiplication Formula
- 35. 35 Example 6 –Solution To approximate the answer, we use a calculator in radian mode and get z1z2 10(–0.2588 + 0.9659i) = –2.588 + 9.659i cont’d
- 36. 36 Example 6 –Solution (b) By the Division Formula cont’d
- 37. 37 Example 6 –Solution Using a calculator in radian mode, we get the approximate answer: cont’d
- 38.
- 39. 39 De Moivre’s Theorem Repeateduse of the Multiplication Formula gives the following useful formula for raising a complex number to a power n for any positive integer n. This theorem says: To take the nth power of a complex number, we take the nth power of the modulus and multiply the argument by n.
- 40. 40 Example 7 –Finding a Power Using De Moivre’s Theorem Find . Solution: Since , it follows from Example 5(a) that So by De Moivre’s Theorem
- 41. 41 Example 7 –Solution cont’d
- 42. 42 nth Roots ofComplex Numbers
- 43. 43 nth Roots ofComplex Numbers An nth root of a complex number z is any complex number w such that wn = z. De Moivre’s Theorem gives us a method for calculating the nth roots of any complex number.
- 44. 44 nth Roots ofComplex Numbers
- 45. 45 Example 8 –Finding Roots of a Complex Number Find the six sixth roots of z = –64, and graph these roots in the complex plane. Solution: In polar form z = 64(cos + i sin ). Applying the formula for nth roots with n = 6, we get for k = 0, 1, 2, 3, 4, 5.
- 46. 46 Example 8 –Solution Using 641/6 = 2, we find that the six sixth roots of –64 are cont’d
- 47. 47 Example 8 –Solution cont’d
- 48. 48 Example 8 –Solution All these points lie on a circle of radius 2, as shown. cont’d The six sixth roots of z = –64
- 49. 49 nth Roots ofComplex Numbers When finding roots of complex numbers, we sometimes write the argument of the complex number in degrees. In this case the nth roots are obtained from the formula for k = 0, 1, 2, . . . , n – 1.