Find the 5 th root of $16 + 16 \sqrt{3} i$ that graphs in the third quadrant.
Convert to polar form. $r (\cos θ + i \sin θ)$
Find $r$ and $θ$
Using $\sin^{- 1}$ , cos-1, or $\tan^{- 1}$ , find $θ$ .
$32 (\cos [?]^{\circ} + i \sin [?]^{\circ})$
Express $θ$ in degrees.

Answer

Expert–verified

Hide Steps

Answer

Find the 5th root in the third quadrant: $32^{\frac{1}{5}} (\cos {(\frac{180 + 60 + 360 k}{5})}^{\circ} + i \sin {(\frac{180 + 60 + 360 k}{5})}^{\circ})$ , for $k = 0, 1, 2, 3, 4$

Steps

Step 1 ：Convert to polar form: $32 (\cos θ + i \sin θ)$

Step 2 ：Find r and $θ$ : $r = \sqrt{16^{2} + (16 \sqrt{3})^{2}} = 32$ and $θ = \tan^{- 1} (\frac{16 \sqrt{3}}{16}) = 60^{\circ}$

Step 3 ：Find the 5th root in the third quadrant: $32^{\frac{1}{5}} (\cos {(\frac{180 + 60 + 360 k}{5})}^{\circ} + i \sin {(\frac{180 + 60 + 360 k}{5})}^{\circ})$ , for $k = 0, 1, 2, 3, 4$

Find the 5 th root of 16+163i that graphs in the third quadrant.Convert to polar form. r(cos⁡θ+isin⁡θ)Find r and θUsing sin−1, cos-1, or tan−1, find θ.32(cos⁡[?]∘+isin⁡[?]∘)Express θ in degrees.

Answer

Popular Problems

Find the 5 th root of $16 + 16 \sqrt{3} i$ that graphs in the third quadrant.
Convert to polar form. $r (\cos θ + i \sin θ)$
Find $r$ and $θ$
Using $\sin^{- 1}$ , cos-1, or $\tan^{- 1}$ , find $θ$ .
$32 (\cos [?]^{\circ} + i \sin [?]^{\circ})$
Express $θ$ in degrees.