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Express 
z_(1)=3-3i in polar form. Express your answer in exact terms, using radians, where your angle is between 0 and 
2pi radians, inclusive.

z_(1)=

Express z1=33i z_{1}=3-3 i in polar form. Express your answer in exact terms, using radians, where your angle is between 00 and 2π 2 \pi radians, inclusive.\newlinez1= z_{1}=

Full solution

Q. Express z1=33i z_{1}=3-3 i in polar form. Express your answer in exact terms, using radians, where your angle is between 00 and 2π 2 \pi radians, inclusive.\newlinez1= z_{1}=
  1. 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 r = \sqrt{a^2 + b^2} , where a a and b b are the real and imaginary parts of the complex number, respectively. For z1=33i z_1 = 3 - 3i , we have a=3 a = 3 and b=3 b = -3 .
  2. Calculating Magnitude: Calculate the magnitude r r using the values of a a and b b :\newliner=32+(3)2=9+9=18=32 r = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} .
  3. Finding Angle: Next, we find the angle θ θ using the formula θ=arctan(ba) θ = \arctan\left(\frac{b}{a}\right) . However, since the complex number is in the third quadrant (both real and imaginary parts are negative), we need to add π π to the angle obtained from the arctan function to get the correct angle in the range 0 0 to 2π radians.
  4. Calculating Angle: Calculate the angle θ θ :\newlineθ=arctan(33)=arctan(1) θ = \arctan\left(\frac{-3}{3}\right) = \arctan(-1) .\newlineThe arctan of 1-1 is π4 -\frac{π}{4} , but since the complex number is in the third quadrant, we add π π to get the correct angle:\newlineθ=π4+π=3π4 θ = -\frac{π}{4} + π = \frac{3π}{4} .
  5. Correcting Angle Calculation: However, there is a mistake in the previous step. The angle 3π4 \frac{3π}{4} corresponds to the second quadrant, not the third. For the third quadrant, we should add π π to π4 \frac{π}{4} , not subtract it. Let's correct this:\newlineθ=π4+π=5π4 θ = \frac{π}{4} + π = \frac{5π}{4} .

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