Express the complex number 𝑧 is equal to four 𝑖 in trigonometric form.
𝑧 is equal to 𝑎 plus 𝑏 𝑖 as known as the rectangular form of the complex number 𝑧. If we compare this form to our complex number 𝑧 is equal to four 𝑖, we can see that 𝑎 must be equal to zero and 𝑏 has a value of four since 𝑏 is the coefficient of 𝑖. When we write a complex number in trigonometric or polar form, we write it as 𝑧 is equal to 𝑟 multiplied by cos 𝜃 plus 𝑖 sin 𝜃, where 𝑟 is known as the modulus of the complex number 𝑧 and 𝜃 is the argument.
In polar form, 𝜃 can be in degrees or radians. The radians is often preferred, whereas in exponential form it does need to be in radians. So we need to find a way to represent the real and complex components of our number in terms of 𝑟 and 𝜃. In fact, we can use this formula to help us. The modulus 𝑟 is the square root of 𝑎 squared plus 𝑏 squared. This is derived from the Pythagorean theorem. And to find 𝜃, we can use tan 𝜃 is equal to 𝑏 over 𝑎.
So let’s substitute what we know about our complex number into these formula: 𝑟 is equal to the square root of 𝑎 squared plus 𝑏 squared, which is the square root of zero squared plus four squared, which is simply four. Tan 𝜃 is equal to four divided by zero. Now this is actually undefined. However, we do know that the tangent function is undefined at 𝜋 over two and then at intervals of two 𝜋. So 𝜃 must be 𝜋 over two for this to be true.
Now that we have values for 𝑟 and 𝜃, let’s substitute them into the formula for the trigonometric form of the complex number. Doing so and we can see that 𝑧 is equal to four multiplied by cos of 𝜋 over two plus 𝑖 sin 𝜋 over two.