Solve. $$ x^{4}+16 i=0 $$ The complex solutions in trigonometric form are (Simplify your answers, including any radicals. Use integers or decimals for any numbers in the expression. Type any angle measures in degrees. Do not include the degree symbol in your answer. Use a comma to separate answers as needed.)

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We start with the equation
$$
x^4+16i=0.
$$
First, rewrite it as
$$
x^4=-16i.
$$

**Step 1. Express the right‐hand side in trigonometric form**

Recall that any complex number $$z = a+bi$$ can be written in trigonometric (polar) form as
$$
z = r(\cos\theta+i\sin\theta),
$$
where
$$
r=\sqrt{a^2+b^2} \quad \text{and} \quad \theta=\tan^{-1}\left(\frac{b}{a}\right).
$$

For $$z=-16i$$, we have:
$$
a=0,\quad b=-16.
$$
Thus, the modulus is
$$
r=\sqrt{0^2+(-16)^2}=\sqrt{256}=16,
$$
and the argument is
$$
\theta=-90^\circ \quad \text{(or equivalently }270^\circ\text{)}.
$$
So we can write
$$
-16i=16\left(\cos(-90^\circ)+i\sin(-90^\circ)\right).
$$

**Step 2. Write $$x$$ in polar form and take the fourth roots**

Let
$$
x= r_x\left(\cos\phi+i\sin\phi\right).
$$
Since
$$
x^4=r_x^4\left(\cos4\phi+i\sin4\phi\right),
$$
we need
$$
r_x^4=16 \quad \text{and} \quad 4\phi=-90^\circ+360^\circ k,\quad k=0,1,2,3.
$$

Solve for $$r_x$$:
$$
r_x^4=16 \quad \Longrightarrow \quad r_x=\sqrt[4]{16}=2.
$$

Solve for $$\phi$$:
$$
4\phi=-90^\circ+360^\circ k \quad \Longrightarrow \quad \phi=\frac{-90^\circ+360^\circ k}{4}.
$$

**Step 3. Write the solutions**

For $$k=0,1,2,3$$, we obtain:

- For $$k=0$$:
  $$
  \phi=\frac{-90^\circ+0}{4}=-22.5^\circ.
  $$
- For $$k=1$$:
  $$
  \phi=\frac{-90^\circ+360^\circ}{4}=\frac{270^\circ}{4}=67.5^\circ.
  $$
- For $$k=2$$:
  $$
  \phi=\frac{-90^\circ+720^\circ}{4}=\frac{630^\circ}{4}=157.5^\circ.
  $$
- For $$k=3$$:
  $$
  \phi=\frac{-90^\circ+1080^\circ}{4}=\frac{990^\circ}{4}=247.5^\circ.
  $$

Thus, the four solutions are:
$$
x=2\left(\cos(-22.5^\circ)+i\sin(-22.5^\circ)\right),\quad 2\left(\cos(67.5^\circ)+i\sin(67.5^\circ)\right),\quad 2\left(\cos(157.5^\circ)+i\sin(157.5^\circ)\right),\quad 2\left(\cos(247.5^\circ)+i\sin(247.5^\circ)\right).
$$

These are the complex solutions in trigonometric form.
The complex solutions in trigonometric form are: $$ x = 2\left(\cos(-22.5^\circ) + i\sin(-22.5^\circ)\right), \quad 2\left(\cos(67.5^\circ) + i\sin(67.5^\circ)\right), \quad 2\left(\cos(157.5^\circ) + i\sin(157.5^\circ)\right), \quad 2\left(\cos(247.5^\circ) + i\sin(247.5^\circ)\right). $$

Solve.

The complex solutions in trigonometric form are
(Simplify your answers, including any radicals. Use integers or decimals for any
numbers in the expression. Type any angle measures in degrees. Do not include the
degree symbol in your answer. Use a comma to separate answers as needed.)

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Solve. The complex solutions in trigonometric form are (Simplify your answers, including any radicals. Use integers or decimals for any numbers in the expression. Type any angle measures in degrees. Do not include the degree symbol in your answer. Use a comma to separate answers as needed.)