Given that 𝑧 one is equal to four multiplied by cos 45 degrees plus 𝑖 sin of 45 degrees and 𝑧 two is equal to six of cos of 90 degrees plus 𝑖 sin of 90 degrees, find the exponential form of 𝑧 two over 𝑧 one.
There are two ways that we can attempt this problem. We’re given two complex numbers in trigonometric form. And we’re told to find their quotient and represent that in exponential form. We could begin by converting both of our numbers to exponential form first and then evaluating 𝑧 one over 𝑧 two. Or we could recall the rules for dividing two complex numbers in trigonometric form and divide our numbers first.
Let’s consider both methods. For complex number of the form 𝑟 cos 𝜃 plus 𝑖 sin 𝜃, we write it in exponential form as 𝑟𝑒 to the 𝑖𝜃. Now, it’s important to remember that to do this, our argument 𝜃 must be in radians. And here, we noticed that throughout we’ve been given the argument of our two complex numbers in degrees. So how do we change a number from degrees to radians. Well, there is a rule. But let’s recall where it comes from.
We know that two 𝜋 radians is equal to 360 degrees. We can find the value of one degree by dividing through by 360. And when we do, we see that one degree is equal to two 𝜋 by 360 radians. And since that simplifies to 𝜋 over 180, we can change from degrees to radians by multiplying by 𝜋 by 180. We’re given that the argument for our first complex number is 45 degrees. So we need to begin by converting 45 degrees into radians. And as we said before, we do that by multiplying it by 𝜋 over 180. That’s equal to 𝜋 by four radians.
And while we’re here, let’s convert the argument for the second complex number, 𝑧 two, into radians. It’s 90 multiplied by 𝜋 by 180 which is 𝜋 by two. And now that we know the argument for both our complex numbers in radians, we can use the conversion formula. The modulus for our first complex number is four. So we can see that, in exponential form, 𝑧 one can be written as four 𝑒 to the 𝜋 by four 𝑖. And for our second complex number, its modulus is six. So we can say that 𝑧 two in exponential form is six 𝑒 to the 𝜋 by two 𝑖.
And to find their quotient, we can apply the general rules of indices. We’re dividing six 𝑒 to the 𝜋 by two 𝑖 by four 𝑒 to the 𝜋 by four 𝑖. And we recall that when we divide two numbers with exponents when their bases are the same, we simply subtract the exponents. So 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 minus 𝑏. And since we know that one-half minus one-quarter is one-quarter, we can say that 𝜋 by two minus 𝜋 by four is 𝜋 by four. And then we simplify six over four. And it becomes three over two. So we can see that the exponential form of 𝑧 one over 𝑧 two is three over two multiplied by 𝑒 to the 𝜋 by four 𝑖.
Now, let’s consider the second method. And that was to use what we know about dividing complex numbers in trigonometric form to work out the quotient before converting it. Here, to divide complex numbers in trigonometric form, we divide their moduli. And we subtract their arguments. And at this point, you should begin to see the relationship between the two methods we’re applying. When we divide their moduli, that’s the 𝑟 values, that’s six divided by four which simplifies to three over two.
And we can work out the argument for our quotient by subtracting the argument of 𝑧 one from the argument of 𝑧 two. That’s 𝜋 by two minus 𝜋 by four. And once again, 𝜋 by two minus 𝜋 by four is 𝜋 by four. So in trigonometric form, our quotient is three over two multiplied by cos of 𝜋 by four plus 𝑖 sin 𝜋 by four. And once again, we can use our conversion formulae to convert this from trigonometric form into exponential form. This time its modulus is three over two. And its argument is 𝜋 over four. And once again, we see that 𝑧 one over 𝑧 two in exponential form is three over two multiplied by 𝑒 to the 𝜋 by four 𝑖.