in part two of our lesson on the trig form of complex numbers you will learn how to convert from rectangular form to trig form in this example, we have to express in complex trig form z is equal to two radical three minus 2i so let's first graph this number in the complex coordinate plane we know that the x value is two radical three, so that's a positive and the y value
is -2 so that means our vector goes here this will be r and this will be theta using our conversion factors we know that r is equal to the square root of x squared plus y squared so that will be the square root
of two radical three squared plus a -2 squared which is equal to the square root we know that two radical three squared becomes four times three which is 12 plus -2 squared
is four this is the square root of 16 which is four, so we know r is four the tangent of theta is equal to y over x so in our case that is -2 divided by two radical three
which is equal to -1 over radical three which we can rationalize multiply by radical three and the tangent of theta becomes negative radical 3 over 3 now from our special angle chart we know that the reference angle is going to be pi over 6
or if we're in degrees then the reference angle would be 30â° now we have to be in quadrant four so if our reference angle here is pi over 6 then we know that theta in quadrant four has to be two pi minus pi over 6 which is 11 pi over 6 then theta
is 360â° minus 30â° which is 330â° we know that the trig form is z equals r cis theta so if we are in radians then z is going to equal four cic 11 pi over 6 or if we are in degrees
then z would equal four cis 330â° z is equal to -4 plus 4i it's cyu time, so pause the video work the example on your own then restart the video to check your answer ok let's see how you did so the real part is -4
the imaginary part is positive four here is z this will be r, and this will be theta using our conversions we know that r is going to equal the square root of -4 squared plus four squared
16 plus 16 is 32 we know that 32 has a perfect square of 16 in it, and the irreducible part is two so r is equal to four radical two we know that tangent of theta is equal to y over x the y value is four x is -4 this reduces to
-1 referring to our special angle chart we know that theta r is equal to pi over 4 in radians or 45â° theta is in quadrant two so if we put the reference angle here in our picture
start at pi, and we back up by pi over 4 then we know that we would have theta equal to 3 pi over four we would start at 180 and back up by 45, which means we are at 135â° trig form means that z has to be expressed as r cis theta will equal four radical two cis
if we are in degrees then z will equal four radical two 135 degrees in this example, we will express in complex trig form z is equal to two i so this means that you could rewrite this as z is equal to zero plus two i so then you know that x would be zero
and y is equal to two is here we know that r has to be the square root which is the square root of zero squared plus two squared which is the square root of four which is two you could've also seen that from the beginning
knowing that r of course is the distance from the origin out to the terminal point of the vector likewise it's pretty easy to see what theta is in the picture without having to do a calculation but i want you to see that it would work out even if you did do the calculation we know tangent theta which is 2/0 which is undefined we know that looking at our picture seeing that theta has to be on the positive y axis
we know that theta then if we're in radians would be pi over 2 theta is 90â° we have to write z in trig form which we know is z equals r cis theta z is equal to two cis pi over 2 and if we are in degrees z will equal two
90â° z is equal to -2 let's go ahead and check your understanding on this one so pause the video, work the example on your own we can write this in standard form by setting it equal to -2 plus zero i so we know that our x value and our y value is zero so then here is our vector
this distance will be r, this angle is theta and this is one of those cases where you can see r and theta immediately but if for some reason you don't or you're not sure just apply the definitions which equals the square root of -2 squared zero squared
and notice that r is always positive sometimes students will make mistakes and think that r is negative two because it's going in this direction remember that r is still a distance. it's the distance from the origin out to the terminal point and that distance r is always positive using the definitions, tangent of theta is equal to y over x, we know our y value is zero x is -2 zero divided by -2 is zero
and since theta has to be on the negative x axis we know that theta is equal to pi if we're in the radian measurement system then theta will equal 180 degrees z will equal two cis pi and if we're in degrees cis 180 degrees in this example, we have to express z is equal to one plus five i in complex trig form we will round our radian answers to hundredths and degree answers to the tenths place
so this will be a calculator problem if you want to go ahead and get your calculators out first we will want to graph z we know that the real part is one and the imaginary part is five so here's the vector that is z we have to find r and theta using our definitions we now r is equal to the square root of x squared plus y squared
so that's going to be the square root of one squared plus five squared of 26 so now we know what r is tangent of theta is y over x so that's going to be five divided by one which is equal to five
now if we take the tangent inverse of both sides so we have tangent inverse of tangent theta is going to equal the tangent inverse of five we get that theta is going to equal now we need to make sure that our calculator is in radians mode, so we hit the drg button, cursor under the radians and hit enter now we do second tangent inverse of five and rounding to the hundredths place we see that theta would be 1.37
we put our calculator into degree mode, so go back to the drg button cursor under degrees and hit enter we've got tangent inverse of five and rounding to the tenths place we get 78.7â° we know is z equals r cis theta so if we're in radians we would say z is equal to radical 26 cis 1.37
we will say that z is equal to radical 26 78.7â°
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