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All right in this video we×39’re going to-do an example of expressing a complex number in trigonometric or polar Normand the idea is okay we#39’ve got some complex number Z we can write that as Plus y I if the real part is X and the imaginary part is y so this complex number we can make an angle with the positive x-axis and that will simply be our angle theta if you think about the distance out to that point the distance after the point is the modulus, but we're going to call that R can kind of think about that as being a radius Ina sense well we could come up with some relationships we could do cosine, so cosine would be cosine of theta would be adjacent x over the hypotenuse which ISR if you solve that for X we'll get equals R cosine theta the same way we could solve for y get y equals R sine theta if you replace the X with R cosine theta and the y with our sine theta toucan factor the r out, and it basically says we can rewrite our complex number Was R times the quantity cosine theta plus I sine thetas let's try to make some sense out ozone of these suppose we want to we×39;redoing to convert this complex number to plus 2i into polar form okay so the first thing IN#39’m going to do is just make little sketch here so over to up -okay so their×39’s our there's our complex number Z equals 2plus 2i so now to go back to our formulas again the first thing I'm going to figure out is R which is our modulus so R is just going to be the modulus, and you get the modulus we just take each part the real part and the imaginary part, and we square them add them together in square root so we×39’ll get 4plus 4 which is 8 so with square root of8 we can rewrite that though is the square root of 4 times the square root of 2 which will give us 2 root 2 so now×39’ve got our R value well we don't know our angle theta here just yet, but we configure that out just using again little trigonometry, so the adjacent side has length to the opposite side also has a length of 2 if we do tangent of theta we would get opposite over adjacent so 2 over 2 which is going to be 1 well tangent of PI over 4 or 45degreesthat'’ll give us, solo now we know our angle theta and now we×39’re in business pecan rewrite our complex number it says we write R so the R value we said was 2root 2, and then we simply take cosine ofthetawhich is kind of sign up 45 degrees plus times' sine of 45 degrees and now thesis what we×39’ve got what we want this would be our final solution here okay×39’ve now taken our complex number and put it in polar form
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