Find the 5 th root of \( 16+16 \sqrt{3} i \) that graphs in the third quadrant.
Convert to polar form. \( r(\cos \theta+i \sin \theta) \)
Find \( r \) and \( \theta \)
Using \( \sin ^{-1} \), cos-1, or \( \tan ^{-1} \), find \( \theta \).
\[
32\left(\cos [?]^{\circ}+i \sin [?]^{\circ}\right)
\]
Express \( \theta \) in degrees.
Answer
Find the 5th root in the third quadrant: \( 32^{\frac{1}{5}}\left(\cos \left(\frac{180+60+360k}{5}\right)^{\circ} + i \sin \left(\frac{180+60+360k}{5}\right)^{\circ}\right) \), for \( k=0,1,2,3,4 \)
Step 1 ：Convert to polar form: \( 32(\cos \theta + i \sin \theta) \)
Step 2 ：Find r and \( \theta \): \( r = \sqrt{16^2 + (16\sqrt{3})^2} = 32 \) and \( \theta = \tan^{-1}(\frac{16\sqrt{3}}{16}) = 60^{\circ} \)
Step 3 ：Find the 5th root in the third quadrant: \( 32^{\frac{1}{5}}\left(\cos \left(\frac{180+60+360k}{5}\right)^{\circ} + i \sin \left(\frac{180+60+360k}{5}\right)^{\circ}\right) \), for \( k=0,1,2,3,4 \)
