Question

Write the complex number in Trig form r(cos+isin), with the interval [0deg, 360deg)
-4+4isqrt3

Answer 1

\[\begin{cases}r\cos\theta=-4\\
r\sin\theta=4\sqrt3\end{cases}\]
\[r^2=r^2(\cos^2\theta+\sin^2\theta)=(-4)^2+(4\sqrt3)^2~~\iff~~r^2=64~~\iff~~r=8\]
So then,
\[r\cos\theta=-4~~\iff~~8\cos\theta=-4~~\iff~~\cos\theta=-\frac{1}{2}\]
and
\[r\sin\theta=4\sqrt3~~\iff~~8\sin\theta=4\sqrt3~~\iff~~\sin\theta=\frac{\sqrt3}{2}\]
What angle \(\theta\) satisfies these ratios?

Answer 2

120?

Answer 3

so i would write 120deg -1/2 sqrt3/2?

Answer 4

You're right about the angle, but the trig form would be \(r(\cos\theta+i\sin\theta\), or \(8\cos120^\circ+8i\sin120^\circ\).

Answer 5

and that's just the answer?