This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions for completing the table of complex numbers: Q2. Complete the following table. | | x+yi | modulus (r) | argument () | polar form | exp form | |---|---|---|---|---|---| | (a) | -2 + 2isqrt(3) | 4 | (2)/(3) | 4( (2)/(3) + i (2)/(3)) | 4e^i(2)/(3) | | (b) | -3sqrt(2)2 - i3sqrt(2)2 | 3 | -(3)/(4) | 3( (-(3)/(4)) + i (-(3)/(4))) | 3e^-i(3)/(4) | | (c) | 3sqrt(3) + 3i | 6 | ()/(6) | 6( ()/(6) + i ()/(6)) | 6e^i()/(6) | | (d) | i | | ()/(2) | ( ()/(2) + i ()/(2)) | e^i()/(2) | Here are the step-by-step calculations for each row: a) Given x+yi = -2 + 2isqrt(3) Step 1: Calculate the modulus r. r = sqrt(x^2 + y^2) = sqrt((-2)^2 + (23))^2 = sqrt(4 + 12) = sqrt(16) = 4 Step 2: Calculate the argument . Since x < 0 and y > 0, the complex number is in the second quadrant. = |(y)/(x)| = |2sqrt(3)-2| = sqrt(3) = ()/(3) = - = - ()/(3) = (2)/(3) Step 3: Write the polar form. r( + i ) = 4( (2)/(3) + i (2)/(3)) Step 4: Write the exponential form. re^i = 4e^i(2)/(3) b) Given modulus r = 3 and argument = -(3)/(4) Step 1: Write the polar form. r( + i ) = 3( (-(3)/(4)) + i (-(3)/(4))) Step 2: Calculate x+yi. x = r = 3 (-(3)/(4)) = 3 (-sqrt(2)2) = -3sqrt(2)2 y = r = 3 (-(3)/(4)) = 3 (-sqrt(2)2) = -3sqrt(2)2 x+yi = -3sqrt(2)2 - i3sqrt(2)2 Step 3: Write the exponential form. re^i = 3e^-i(3)/(4) c) Given polar form 6( ()/(6) + i ()/(6)) Step 1: Identify the modulus and argument from the polar form. r = 6 = ()/(6) Step 2: Calculate x+yi. x = r = 6 (()/(6)) = 6 (sqrt(3)2) = 3sqrt(3) y = r = 6 (()/(6)) = 6 ((1)/(2)) = 3 x+yi = 3sqrt(3) + 3i Step 3: Write the exponential form. re^i = 6e^i()/(6) d) Given exponential form e^i()/(2) Step 1: Identify the modulus and argument from the exponential form. r = = ()/(2) Step 2: Write the polar form. r( + i ) = ( ()/(2) + i ()/(2)) Step 3: Calculate x+yi. x = r = (()/(2)) = · 0 = 0 y = r = (()/(2)) = · 1 = x+yi = i That's 4 down. 1 left today — send the next one.