Question 118071This question is from textbook Advanced Mathematics - precalculus with discrete mathamatics and data analysis
: find the square roots of 3+4i
This question is from textbook Advanced Mathematics - precalculus with discrete mathamatics and data analysis
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Here's a way you can do this problem.
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Plot 3 + 4i on a coordinate system by using 3 as the value on the x axis and from that point
(the point 3, 0) go vertically 4 units. You should now be at the (x, y) point (3, 4). Draw
a line from the origin to the point (3, 4). Notice that you have formed a right triangle
consisting of 1 leg 3 units long along the x-axis; one leg 4 units long vertically between the
points (3,0) and (3,4); and a hypotenuse connecting the origin (0,0) with the point (3,4).
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Using the Pythagorean theorem you can find the magnitude of the hypotenuse (H) through the
relationship that says the sum of the squares of the two legs equals the hypotenuse
squared. In equation form this is:
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(L1)^2 + (L2)^2 = H^2
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Call L1 the leg along the x-axis and it is 3 units long. Call L2 the vertical leg and it
is 4 units long. So the equation becomes:
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3^2 + 4^2 = H^2
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3^2 = 9 and 4^2 = 16 so the equation reduces to:
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9 + 16 = 25 = H^2
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Taking the square root of both sides results in H = 5. So the magnitude of the polar form is 5.
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The angle of the polar form is the angle between the x-axis and the hypotenuse. It can be
found by recognizing that the tangent of that angle is opposite/adjacent = 4/3. If you take the
arctangent of 4/3 you get the angle as 53.13010235 degrees.
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So now you have the polar form of 3 + 4i as being 5 at 53.13010235 degrees. To find the
square root of the polar form you do two things. One, you take the square root of the magnitude
of the polar form. In this case you take the square root of 5 and you get two values ...
+2.236067977 and -2.236067977. Two you take half the value of the angle ... and half of
53.13010235 is 26.56505118 degrees. So the two answers to this problem in polar form (magnitude
and angle) are +2.236067977 at 26.56505118 degrees and -2.236067977 at 26.56505118 degrees.
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But you probably need the answers to be in rectangular form (real part which is horizontal
and imaginary part which is vertical ... as an example 3 + 4i is rectangular form).
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You can convert you answers to rectangular form using the fact that the horizontal leg
equals the hypotenuse times the cosine of 26.56505118 degrees, and the vertical or imaginary
leg equals the hypotenuse times the sine of 26.56505118 degrees. Let's do that for both answers.
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First when the hypotenuse is +5 we get that the horizontal (real) part is +5 * cosine(26.56505118).
Do the multiplication and you get an answer of +2. And the associated vertical (imaginary) part
is +5 * sine(26.56505118) and when you do the multiplication this time you get 1. This tells
you that the rectangular form of one of the answers is 2 + 1i or just 2 + i in conventional form.
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Then, when the hypotenuse is -5 we get that the horizontal (real) part is -5 * cosine(26.56505118).
Do the multiplication and you get an answer of -2. And the associated vertical (imaginary) part
is -5 * sine(26.56505118) and when you do the multiplication this time you get -1. This tells
you that the rectangular form of the second answer is -2 - 1i or just -2 - i in conventional form.
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In summary, the two answers for the square root of 3 + 4i are 2 + i and -2 - i.
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Let's check by squaring each of the two answers:
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First (2 + i)*(2 + i) = 2^2 + 2i + 2i + (i)^2
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2^2 = 4 .... 2i + 2i = 4i ... and, by definition, (i)^2 = -1
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Substitute these values and the equation reduces to:
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(2 + i)*(2 + i) = 4 + 4i - 1
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and finally, combining the +4 and the -1 reduces it to 3 + 4i. So, when we squared
2 + i we got 3 + 4i ... and therefore, 2 + i is a square root of 3 + 4i
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Next (-2 - i)*(-2 - i) = (-2)^2 + 2i + 2i + (i)^2
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(-2)^2 = +4 .... 2i + 2i = 4i ... and again , by definition, (i)^2 = -1
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Substitute these values and the equation reduces to:
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(-2 - i)*(-2 - i) = 4 + 4i - 1
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and finally, combining the +4 and the -1 reduces it to 3 + 4i. So, when we squared
-2 - i we got 3 + 4i ... and therefore, -2 - i is also a square root of 3 + 4i
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Hope this doesn't confuse you too much. Stick with it and try to understand each process and step in
reaching this answer. An the two answers are correct ... but if your text is only interested
in a principal answer, then that answer would be 2 + i.
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