Complex Numbers Chapter 2 66 repeatedly, and find exactly 4 fourth roots. We can read the values of 64 in rectangular form from Figure 10.3: V-64=+2 ± 2i (all four combinations of ± signs) or we can calculate them as in Example 2, or we can solve the equation 24 = -64 by computer Example 4. Find and plot all values of -81. The polar coordinates of -8i are r = 8, 0 270°360°k = 3T/2 2Tk. Then the polar coordinates of -8i are 270+360°k = 45°+60°k or (10.5) In Figure 10.4, we sketch a circle of radius V2. On it we plot the point at 45° and then plot the rest of the 6 equally spaced points 60° apart. To find the roots in rectangular coordinates, we need to find all the values of r(cos 0 i sin 0) with r and 0 given by (10.5). We can do this one root at a time or more simply by using a computer to solve the equation 26 = -8i. We find (see Problem 33) Figure 10.4 1 1.366 -0.366i, 0.366-1.366i}. Summary In each of the preceding examples, our steps in finding Vrei were (a) Find the polar coordinates of the roots: Take the nth root of r and divide e+2kT by n. (b) Make a sketch: Draw a circle of radius, plot the root with angle 0/n, and then plot the rest of the n roots around the circle equally spaced 2T/n apart. Note that we have now essentially solved the problem. From the sketch you can see the approximate rectangular coordinates of the roots and check your answers in (c). Since this sketch is quick and easy to do, it is worthwhile even if you use a computer to do part (c) (c) Find the r+iy coordinates of the roots by one of the methods in the examples. If you are using a computer, you may want to make a computer plot of the roots which should be a perfected copy of your sketch in (b). PROBLEMS, SECTION 10 Follow steps (a), (b), (c) above to find all the values of the indicated roots VT 2. 27 VT 1. 3. VI6 5. 4. 6. 7. 32 11. S 10. 12. Section 11 The Exponential and Trigonometric Functions 67 64 14 13. 15. 16 17 19 2+2iv3 20. 21
Complex Numbers Chapter 2 66 repeatedly, and find exactly 4 fourth roots. We can read the values of 64 in rectangular form from Figure 10.3: V-64=+2 ± 2i (all four combinations of ± signs) or we can calculate them as in Example 2, or we can solve the equation 24 = -64 by computer Example 4. Find and plot all values of -81. The polar coordinates of -8i are r = 8, 0 270°360°k = 3T/2 2Tk. Then the polar coordinates of -8i are 270+360°k = 45°+60°k or (10.5) In Figure 10.4, we sketch a circle of radius V2. On it we plot the point at 45° and then plot the rest of the 6 equally spaced points 60° apart. To find the roots in rectangular coordinates, we need to find all the values of r(cos 0 i sin 0) with r and 0 given by (10.5). We can do this one root at a time or more simply by using a computer to solve the equation 26 = -8i. We find (see Problem 33) Figure 10.4 1 1.366 -0.366i, 0.366-1.366i}. Summary In each of the preceding examples, our steps in finding Vrei were (a) Find the polar coordinates of the roots: Take the nth root of r and divide e+2kT by n. (b) Make a sketch: Draw a circle of radius, plot the root with angle 0/n, and then plot the rest of the n roots around the circle equally spaced 2T/n apart. Note that we have now essentially solved the problem. From the sketch you can see the approximate rectangular coordinates of the roots and check your answers in (c). Since this sketch is quick and easy to do, it is worthwhile even if you use a computer to do part (c) (c) Find the r+iy coordinates of the roots by one of the methods in the examples. If you are using a computer, you may want to make a computer plot of the roots which should be a perfected copy of your sketch in (b). PROBLEMS, SECTION 10 Follow steps (a), (b), (c) above to find all the values of the indicated roots VT 2. 27 VT 1. 3. VI6 5. 4. 6. 7. 32 11. S 10. 12. Section 11 The Exponential and Trigonometric Functions 67 64 14 13. 15. 16 17 19 2+2iv3 20. 21
Advanced Engineering Mathematics
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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Problems Section 10 number 9. Please answer the questions
Transcribed Image Text:Complex Numbers
Chapter 2
66
repeatedly, and find exactly 4 fourth roots. We can read the values of 64 in
rectangular form from Figure 10.3:
V-64=+2 ± 2i (all four combinations of ± signs)
or we can calculate them as in Example 2, or we can solve the equation 24 = -64
by computer
Example 4. Find and plot all values of -81. The polar coordinates of -8i are r = 8,
0 270°360°k = 3T/2 2Tk. Then the polar coordinates of -8i are
270+360°k
= 45°+60°k or
(10.5)
In Figure 10.4, we sketch a circle of radius V2. On
it we plot the point at 45° and then plot the rest of
the 6 equally spaced points 60° apart. To find the
roots in rectangular coordinates, we need to find all
the values of r(cos 0 i sin 0) with r and 0 given by
(10.5). We can do this one root at a time or more
simply by using a computer to solve the equation
26 = -8i. We find (see Problem 33)
Figure 10.4
1
1.366 -0.366i, 0.366-1.366i}.
Summary In each of the preceding examples, our steps in finding Vrei were
(a) Find the polar coordinates of the roots: Take the nth root of r and divide
e+2kT by n.
(b) Make a sketch: Draw a circle of radius, plot the root with angle 0/n, and
then plot the rest of the n roots around the circle equally spaced 2T/n apart.
Note that we have now essentially solved the problem. From the sketch you
can see the approximate rectangular coordinates of the roots and check your
answers in (c). Since this sketch is quick and easy to do, it is worthwhile even
if you use a computer to do part (c)
(c) Find the r+iy coordinates of the roots by one of the methods in the examples.
If you are using a computer, you may want to make a computer plot of the
roots which should be a perfected copy of your sketch in (b).
PROBLEMS, SECTION 10
Follow steps (a), (b), (c) above to find all the values of the indicated roots
VT
2. 27
VT
1.
3.
VI6
5.
4.
6.
7.
32
11. S
10.
12.
Section 11
The Exponential and Trigonometric Functions
67
64
14
13.
15.
16
17
19
2+2iv3
20.
21
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