Find the indicated roots. Write the answer in trigonometric form.
-Fourth roots of 256(cos 320° + i sin 320°)
A) 4(cos 320° + i sin 320°), 4(cos 410° + i sin 410°), 4(cos 500° + i sin 500°), 4(cos 230° + i sin 230°)
B) 4(cos 80° + i sin 80°), 4(cos 170° + i sin 170°), 4(cos 260° + i sin 260°), 4(cos 350° + i sin 350°)
C) 4(cos 80° + i sin 80°), 4(cos 170° + i sin 170°), 4(cos 340° + i sin 340°), 4(cos 70° + i sin 70°)
D) 4(cos 160° + i sin 160°), 4(cos 250° + i sin 250°), 4(cos 340° + i sin 340°), 4(cos 70° + i sin 70°)

Question

Quizplus · Accepted Answer

To find the fourth roots of $256(\cos 320° + i \sin 320°)$, we first find the magnitude of the root, which is the fourth root of 256, equal to 4. Then, we divide the angle by 4 and add $k \cdot \frac{360°}{4}$ for $k = 0, 1, 2, 3$ to find all fourth roots. Starting with $320° / 4 = 80°$, and adding $90°$ increments gives angles of $80°$, $170°$, $260°$, and $350°$.

Find the Indicated Roots