Find the indicated roots. Write the answer in trigonometric form.
-Cube roots of 125(cos 276° + i sin 276°)
A) 5(cos 92° + i sin 92°), 5(cos 152° + i sin 152°), 5(cos 32° + i sin 32°)
B) 5(cos 182° + i sin 182°), 5(cos 242° + i sin 242°), 5(cos 122° + i sin 122°)
C) 5(cos 92° + i sin 92°), 5(cos 212° + i sin 212°), 5(cos 212° + i sin 212°)
D) 5(cos 92° + i sin 92°), 5(cos 212° + i sin 212°), 5(cos 332° + i sin 332°)

Question

Quizplus · Accepted Answer

To find the cube roots of a complex number in trigonometric form, $z = r(\cos \theta + i\sin \theta)$, we use the formula for the nth roots: $z_k = \sqrt[n]{r}(\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))$, where $k = 0, 1, ..., n-1$. For cube roots, $n = 3$, and $r = 125$, $\theta = 276°$. Thus, $\sqrt[3]{125} = 5$, and the angles are calculated as $\frac{276° + 2k\pi}{3}$ for $k = 0, 1, 2$. Substituting $k$ values, we get the angles 92°, 212°, and 332°, leading to the roots 5(cos 92° + i sin 92°), 5(cos 212° + i sin 212°), and 5(cos 332° + i sin 332°).

Find the Indicated Roots