benchmarks.wiki

benchmarks.wiki / Public workspace

MATH / The solutions to z⁴=4−4i√(3) can be expressed in the form

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

The solutions to z4=4−4i3z^4 = 4 - 4i \sqrt{3} can be expressed in the form \begin{align*} z_1 &= r_1 (\cos \theta_1 + i \sin \theta_1), \\ z_2 &= r_2 (\cos \theta_2 + i \sin \theta_2), \\ z_3 &= r_3 (\cos \theta_3 + i \sin \theta_3), \\ z_4 &= r_4 (\cos \theta_4 + i \sin \theta_4), \end{align*}where rk>0r_k > 0 and 0∘≤θk<360∘.0^\circ \le \theta_k < 360^\circ. Find θ1+θ2+θ3+θ4,\theta_1 + \theta_2 + \theta_3 + \theta_4, in degrees.
Plain-text mathematical notation (without MathML)
The solutions to z⁴=4−4i√(3) can be expressed in the form
\begin{align*}
z_1 &= r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 &= r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 &= r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 &= r_4 (\cos \theta_4 + i \sin \theta_4),
\end{align*}where r_(k)>0 and 0^(∘)≤θ_(k)<360^(∘).  Find θ₁+θ₂+θ₃+θ₄, in degrees.
Original LaTeX notation
The solutions to $z^4 = 4 - 4i \sqrt{3}$ can be expressed in the form
\begin{align*}
z_1 &= r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 &= r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 &= r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 &= r_4 (\cos \theta_4 + i \sin \theta_4),
\end{align*}where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ.$  Find $\theta_1 + \theta_2 + \theta_3 + \theta_4,$ in degrees.

level

Level 5

type

Precalculus

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
6772a119-27f5-5d0a-ba9f-34ba18968f99
task_key
precalculus--test--6772a119-27f5-5d0a-ba9f-34ba18968f99
task_revision_id
1
config / split
precalculus / test

Official source

initial import