Given that 𝑧 equals three root two multiplied by cos of 225 minus 𝑖 sin of 225, find 𝑧 squared, giving your answer in exponential form.
Euler’s formula says that 𝑒 to the plus or minus 𝑖𝜃 is equal to cos 𝜃 plus or minus 𝑖 sin 𝜃. We can extend this and say that 𝑟 multiplied by 𝑒 to the plus or minus 𝑖 𝜃 is equal to 𝑟 multiplied by cos 𝜃 plus or minus 𝑖 sin 𝜃. We can use this to help us convert between polar and exponential form of a complex number.
The modulus — that’s the value of 𝑟 in our complex number — is three root two. And the argument is given as 225 degrees. However, when we write a complex number in exponential form, we do so using radians. Remember to change a number from degrees to radians, we multiply it by 𝜋 over 180.
So the argument is in fact 225 multiplied by 𝜋 over 180, which is five 𝜋 over four. And since the coefficient of 𝑖 sin 𝜃 is negative, we can rewrite our complex number as three root two multiplied by 𝑒 to the negative five 𝜋 over four 𝑖.
We actually need to work out 𝑧 squared. So we’re going to square this entire expression. 𝑧 squared is, therefore, equal to three root two 𝑒 to the negative five 𝜋 over four 𝑖 all squared. Let’s begin by squaring three root two. Three squared is nine and root two squared is two. So this becomes a nine multiplied by two which is 18. For the exponent, we know that we need to multiply negative five 𝜋 over four 𝑖 by two. And that gives us 𝑒 to the power of negative five 𝜋 over two 𝑖.
And our expression for the complex number 𝑧 squared in exponential form is 18𝑒 to the negative five 𝜋 over two 𝑖. Notice though that each of the exponents in the possible answers to our question have a positive coefficient of 𝑖.
In fact, the imaginary exponential is periodic with a period of two 𝜋. So we can add two 𝜋 to negative five 𝜋 over two repeatedly until we find a positive value. Negative five 𝜋 over two plus two 𝜋 is negative one-half 𝜋. Negative a half of 𝜋 plus two 𝜋 is three 𝜋 over two.
And our expression for 𝑧 squared in exponential form is 18𝑒 to the three 𝜋 over two 𝑖.