Q. Express z1=3−3i in polar form. Express your answer in exact terms, using radians, where your angle is between 0 and 2π radians, inclusive.z1=
Finding Magnitude: To express a complex number in polar form, we need to find its magnitude (r) and angle (θ). The magnitude is found using the formula r=a2+b2, where a and b are the real and imaginary parts of the complex number, respectively. For z1=3−3i, we have a=3 and b=−3.
Calculating Magnitude: Calculate the magnitude r using the values of a and b:r=32+(−3)2=9+9=18=32.
Finding Angle: Next, we find the angle θ using the formula θ=arctan(ab). However, since the complex number is in the third quadrant (both real and imaginary parts are negative), we need to add π to the angle obtained from the arctan function to get the correct angle in the range 0 to 2π radians.
Calculating Angle: Calculate the angle θ:θ=arctan(3−3)=arctan(−1).The arctan of −1 is −4π, but since the complex number is in the third quadrant, we add π to get the correct angle:θ=−4π+π=43π.
Correcting Angle Calculation: However, there is a mistake in the previous step. The angle 43π corresponds to the second quadrant, not the third. For the third quadrant, we should add π to 4π, not subtract it. Let's correct this:θ=4π+π=45π.
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