1 Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working with complex numbers. We will have to go through some basics to get to the point where trigonometric form is most helpful raising complex numbers to powers or finding roots of complex numbers. I will not go into as much detail as a textbook, but try to make sense out of what we are using. We will: Convert standard form of a complex number to trigonometric form. Convert trigonometric form to standard form. Multiply and divide complex numbers. Raise complex numbers to a power using DeMoivre s Theorem. Find all roots to a complex number. Geometry knowledge needed: Basic properties of circles Algebra skills needed: Complex number form and graphing in complex number plane. Multiplying and dividing complex numbers (for checking purposes). Rationaliing a denominator using conjugates Simplifying a radical. Laws of exponents. Trigonometric Form of a Complex Number Recall the basics of complex numbers, where we use the letter. o The standard form of a complex number is a bi, where a and b are real numbers, e.g., +i, -4+i, 0+i, -i, etc. o A complex number has a real part, a, and a pure imaginary part, bi. In the number i, is the real part and -i is the pure imaginary part. o We can graph a complex number in the complex number plane, which has a horiontal real axis and a vertical pure imaginary axis. See the points in the plane in the figure.
2 You can see that this is very much like plotting corresponding points in the rectangular coordinate system. We can work with these complex numbers in much the same way. Let s focus on a general complex number a + bi and bring up some of the ideas that we have been using in this course - r and angle θ. Notice that the complex number is at a distance of r from the origin. Using the right triangle, we know r a b. We use θ and its functions to get cos a b a r cos while sin b r sin r r. Using these relations, we write the standard form of a complex number in trigonometric form. This substitution yields ( r cos ) ( r sin ) i. When we remove parentheses, there can be a little confusion with the imaginary part. The i is not multiplying θ. The definition will rewrite this term and factor the r from both parts. The trigonometric form of the complex number a bi is r(cos i sin ), b where a r cos, b r sin, r a b, and tan. a The r is called the modulus of, and θ is called the argument of.
3 We will take some examples of going back and forth between standard form and trigonometric form, but I want to first point out something that some textbooks do not. Take a good look at the trigonometric form: r(cos i sin ). I have underlined certain important letters. These are used in an abbreviation for the form. It is rcis which is read as r cis θ with the cis pronounced as sis. This is a very handy abbreviation and I will use it. I suppose some authors do not use it for fear you will forget what it means. Example : Convert the number i to trigonometric form. Make a sketch of. The sketch will remind us that θ needs to be in quadrant IV. r a b ( ) 8 We will simplify the radical. r 4 4 Just a reminder of how it is done. b tan a tan ( ) Reference angle is 4, in QIV Then i (cos i sin ). You can write this as cis, but most textbooks will have the longer form as the answers. Some observations: The modulus r is the same thing as finding the absolute value ofa bi. That is, a bi a b r. Absolute value has always been a geometric distance. Like standard form, the trigonometric form is a sum of terms, with the functions determining negative signs. If a special angle (function value) is involved, we will use exact values. I have used degrees here and will continue to do so because they have the most applications and are easier to work with than radians. Your textbook will have some problems with radians or real numbers. Don t forget that when using your calculator for numbers not exact, change the mode to radians. Be careful with the radical sign. Make sure it extends far enough, but don t include any number under it that does not belong. Leave a space between the radical and what comes next if this is a problem. You can check your answer by multiplying out the trigonometric form, e.g., cos? sin?
4 4 Example : Write the complex number 7 4i in trigonometric form. Make a sketch of. We keep in mind that is in quadrant III. r r a b ( 7) ( 4) 6 which cannot be simplified. b 4 4 tan a tan 9.7 as the reference angle. 7 Since is in QIII, Therefore, 74i 6(cos09.7 i sin09.7 ) or 6cis09.7. Example : Write 4(cos0 isin0 ) in standard form. Make a sketch of. We do what the trigonometric form indicates and multiply. Since the reference angle is a special angle of 60, we use exact values. This number is definitely in QII. 4(cos0 isin0 ) 4 i 4 i 4 i The answer above also illustrates what I mentioned about being careful with the radical sign. The i is not under the radical sign. It might be safer to leave a little space after the radical before the i, e.g. i. You can also put the i in front of the term, but that is not usual for standard form. When I write on the board in class, I usually just add a little hook at the end of the radical sign top, as a separator like this: á Now work the problem set for Trigonometric Form of a Complex Number.
5 Problems for Trigonometric Form of a Complex Number. Convert the complex number i to trigonometric form. Make a sketch of.. Write the complex number i in trigonometric form. Make a sketch of.. Write (cos0 isin0 ) in standard form. Make a sketch of.
6 6 Answers to Problems for Trigonometric Form of a Complex Number. The sketch is a reminder that is in quadrant II. r a b r 9 ( ) 8 b tan a tan ( ) Reference angle is 4. Therefore, i (cos i sin ). If radian measure is used, goes in 4 place of.. The sketch is a reminder that is in QIII. r ( ) ( ) 4 r 4 It cannot be simplified. tan = tan 9.0 Reference angle in QI. Therefore, i 4(cos9 i sin9 ).. The sketch reminds us that is in QIV. We will be exact with this special angle. (cos0 i sin0 ) Reference angle i s 0 i( ) i in standard form.
7 7 Multiplication, Division, and Powers of Complex Numbers In algebra, multiplying complex numbers involved the same procedure as multiplying binomials. For example: ( i )( 4 i ) 8 0i i i 8 i ( ) 7i This is not a lengthy process, so using trigonometric form is not necessarily better. It is a step towards where we are heading, however. We can also check some answers this way. Developing the rule for multiplying complex numbers in trigonometric form uses good algebra and past identities: r (cos i sin ) r (cos i sin ) r r ( cos cos i cos sin i sin cos i sin sin ) r r ( cos cos sin sin ) i(cos sin sin cos r r cos( ) i sn i ( In words, the rule tells us to multiply the moduli (plural of modulus) and add the arguments. That requires each complex number to be in trigonometric form. Example : Find the product of (cos60 i sin60 ) and 4(cos0 i sin0 ). This is where the abbreviation is particularly handy. Abbreviate both numbers and multiply. cis60 4cis0 4cis(600 ) 8cis0 Reference angle is 0. 8(cos0i sin0 ) 8( i( )) 4 4i Sometimes you may have to convert each number to trigonometric form first. Also, the sum of the angles or arguments is not always a special angle. When that happens you will have to use your calculator in the appropriate mode and round off as specified. Dividing complex numbers algebraically took considerably more work (Doesn t division always require more effort?). We will look at one example as a reminder. For example, let us divide two complex numbers such as i and i. The process was as follows.
8 8 i i i i i i 6 9i 0i 4 i 6 9i 0( ) 4 ( ) 4 9i 4 9 i Multiply by to rationalie the denominator. Multiplying binomials. Developing the rule for dividing complex numbers in trigonometric form uses this same process along with past identities. The rule: r cos( ) i sin ( ), 0. r In words, to divide two complex numbers in trigonometric form, divide the moduli and subtract the arguments. That certainly seems shorter than algebra, but it does require the trigonometric form first. Example : Find the quotient if (cos0 i sin0 ) and 4(cos70 i sin70 ). Since these are not special angles, to do this problem algebraically would require a lot of decimals. We use the rule to divide (and the handy abbreviation again). cis0 cis(0 70 ) 4cis70 4 cis40 (cos40 i sin40 ) i( ) i 4 4 The real purpose of this section is to get to powers of complex numbers. A power of a number is a repeated multiplication. For example, is a multiplication problem with factors (but only four multiplications). If we want, using the multiplication rule 4 times, then we would have r as a factor times and we would add arguments: r r r r r i cos( ) sin( ) r (cos i sin )
9 9 In general, the power of a complex number is found by: n n r (c osn i s inn), n a positive integer. This is called DeMoivre s Theorem, named after a French mathematician. Example : Find if (cos4 i sin4 ). Using the abbreviation will save a lot of writing and worrying about enough parentheses. (cis 4 ) Using the abbreviation. cis( 4 ) Applying the power rule. 4cis 0 Reference angle is 60 in QII. 4 cos0 i sin0 4 i 4 4 i There is no advantage in writing with mixed numbers. Example 4: Find if i. We will keep in mind that is in quadrant IV. We need to first find the trigonometric form. r tan 00 ( ) 4 tan ( ) Reference angle is 60 in QIV. The trigonometric form of is (cos00 i sin00 ) or cis 00. Raising to a power: (cis00 ) cis(00 ) 8cis 900 Find an equivalent smaller angle. 8cis 80 Subtracted 60 twice. 8(cos80 i sin80) 8( i(0)) 8 0i
10 0 You may have thought of a couple of questions while reading through these examples. Are all of these steps necessary? No. Since I cannot anticipate where someone might have any trouble or a question, I write out every step. The last example, for instance, could be shortened to only the critical steps. r ( ) tan 60 in QI, 00 in QIV cis 00 cis 900 8cis 80 8( i(0)) 8 0i This procedure is still less work than doing it algebraically. What if the angle is not a special angle? In reality, that happens frequently. If directions do not specify, carry the angle you find for trigonometric form out to one decimal place. Since no other calculation is done until the final product or quotient or power, you can then carry out the cosine and sine values to or 4 decimal places, unless otherwise specified. For example, suppose your tangent was a number such that θ =. so that the final trigonometric form for is 4cis.. Then you write out the form and use your calculator. 4cis. 4(cos. i sin. ) 4(.48 i(.9048)).70.69i Depending on the objective, a textbook author may leave the trigonometric form when numbers are like this. If you are raising a complex number to a power, and the parts involve integers only, then the power (final answer) should also be rounded to integers, if that does not automatically happen. á Now work the problem set for Multiplication, Division, and Powers of Complex Numbers.
11 Problems for Multiplication, Division, and Powers of Complex Numbers. Find the product if (cos7 i sin7 ) and (cos i sin ).. Find the quotient if (cos i sin ) and 6(cos i sin ).. Find if (cos0 isin0 ). 4. Find 6 if i.
12 Answers to Multiplying, Dividing, and Powers of a Complex Number cis 7 and cis (cis7 )(cis ) cis(7) 6cis 0 6(cos0 i sin0 ) Reference angle is 0 in QII. 6 i( ) i cis and 6cis cis cis( ) 6cis 6 cis 80 (cos80 i sin80 ) ( i(0)) 0i (cos0 i sin0 ) cis0 cis(0 ) cis 600 Find an equivalent angle less than 60. cis 40 Subtracted 60. (cos40 i sin40 ) Reference angle is 60 in QIII. i( ) 6 6 i i r ( ) ( ) 4 tan cis 4 (cis 4 ) tan 4 and is in QI cis( 64 ) 64cis 70 64(cos70 i sin70 ) 64(0 i( )) 064i
13 Roots of a Complex Number In algebra, you were able to find roots to polynomial equations such as 4 x 0, x 6 0, 8x 0 by factoring, the quadratic formula, or a combination of these methods. You may have also learned how to use theorems to find accurate guesses for roots of higher degree polynomial equations. However, algebra methods failed for finding all solutions to some polynomial equations, such as 7 x 0, a simple equation but not easy to solve. Trigonometry can help find roots to some equations that algebra cannot, or uses an alternate method to solve some of those equations Recall two notations for the n th root of a number a: Using a radical, the notation is n a. n Using exponential form: a. Trigonometry has a method for finding all roots of a simple polynomial equation. It is a result of DeMoivre s Theorem applied in reverse (plus algebra techniques). By definition, a complex number, say w, is the nth root of another complex number, say, if w How do we find the roots? The complex number r(cosi sin ) has exactly n (a positive integer) distinct roots given by: n. cos k n r i sin k where k 0,,,..., n. n n This can look intimidating, so let s look at it in pieces. First, for all roots we take the nth root of the complex number s modulus r. The first root s argument comes from the first π or 60, then dividing by n. The remaining arguments for other roots come from going around the circle (adding π) one more time, before dividing by n, for each argument. Since k starts at 0, k will only go up to n- to get the last root. (Roots repeat after that.)
14 4 Example : Find all the cube roots of. (Let = + 0i.) Write the number in trigonometric form, noting that is on the positive x-axis, and then apply the rule. 0 i r cis 0 cis cis0 (cos0 i sin0 ) ( i 0) 0 60 cis cis 0 i( ) i 0, 0 and tan (60 ) cis cis 40 i( ) i This problem is the equivalent of solving the equation x 0. You can check this by factoring and using the quadratic formula (if you remember how to factor the difference of two cubes). Some textbook authors focus on a shortcut for finding the argument. If you want to do that, it is fine. I think that working neatly and orderly where you can see the thinking process, as in the example, is the best thing to do, plus it is easy to forget shortcuts. Before we go through another example, it is useful to note the geometric significance of the roots. They are spaced equally around the circle. (This example involved special angles, which is not always the case, nor do the roots always start at 0.) While this method is most useful for higher degree (power), I am keeping the example function degrees smaller so as not to overwhelm you at the start. Once you get used to the method, you can (and should) try problems that have more roots.
15 Example : Find all the fourth roots of -4. We first change to trigonometric form, noticing that = i is on the negative x-axis, and then use the rule i r ( 4) 0 4 and tan 80 this time. 4 4cis 80 n r (4) ( ) 80 cis cis 4 i i 4 Simplify the radical cis cis i i 4 80 (60 ) cis cis i( ) i 4 80 (60 ) 4 cis cis i( i 4 Starting at 4 for the first root, we see that the other roots are spaced equally around the circle of radius. See the next figure. This problem is the equivalent of solving the equation x Also, notice that a theorem from algebra that talks about complex roots to a real number occurring in conjugate pairs holds true here. Okay, what happens when there are angles in the roots that are not special? (It is also possible that the trigonometric form of will involve an angle that is not special.)
16 6 Example : Find all fifth roots of = - + i. We notice that is in quadrant II when we write it in trigonometric form. n i r r ( ) and tan. 0 0 Be careful with the double radical. 0 0 cis cis 7.078cis i cis cis cis i 0 (60 ) 0 cis cis 7.078cis i 0 (60 ) 0 4 cis cis 4.078cis i 0 4( 60 ) 0 cis cis.078cis i The roots are equally spaced around a circle of radius 0.078, starting at 7. If you are haphaard about the way you do things, you are apt to make quite a few mistakes. Instead of being in a hurry, go at a steady and orderly pace. á Now work the problem set for Roots of a Complex Number.
17 7 Problems for Roots of a Complex Number. Find all fourth roots of 6. ( = 6 + 0i). Mark the roots on the appropriate circle.. Find all cube roots of -7. ( = i) Mark the roots on the appropriate circle.. Find all cube roots of i. Mark the roots on the appropriate circle.
18 8 Answers to Problems for Roots of Complex Numbers. n 0 i r 6 6cis and tan 0 (on positive x-axis) r cis cis 0 ( i 0) cis cis 90 (0 i()) 0 i 4 0 (60 ) cis cis 80 ( i 0) 4 0 (60 ) 4 cis cis 70 (0 i( ) 0 i 4 You can leave the 0 out in answers and 4.. n 0 i r 7 7cis ( 7) 0 7 and tan 80 ( is on negative x-axis) r 7 (I know it's a lot of 's. 80 cis cis 60 i( ) i cis cis 80 ( i 0) 80 (60 ) cis cis 00 i( ) i )
19 9. n i r ( ) 8 and tan ( is in QIV) 8cis r 8 ( 8) ( ) ( ) ( ) cis cis0 (cos0 i sin0 ) i ( 60 ) cis cis (cos i sin ) i (60 ) cis cis 4 (cos4 i sin4 ) i I hope that this course has been helpful, and that you understand what Trigonometry does. Thank you for giving it a try. You can still contact me if you need help. A Note about the Polar Coordinate System If you are taking Trigonometry in a Pre-Calculus course, the Polar Coordinate System will be a topic, probably a chapter, in your textbook. That is because the Polar Coordinate System is used at times in Calculus. If you are headed toward Calculus, you should do next week s lesson on the Polar Coordinate System.
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