Pergunta
![32(cos60^circ +isin60^circ )
Next, find Theta 's for all 5 roots of 16+16sqrt (3)i
Theta =60^circ ,420^circ ,[?]^circ ,square ^circ ,square ^circ
Remember, to find the different Theta 's add
360^circ to Theta](https://static.questionai.br.com/resource%2Fqaiseoimg%2F202503%2F32cos60circ-isin60circ-next-find-theta-s-5-roots-tTa0PotWaj0w.jpg?x-oss-process=image/resize,w_558,h_500/quality,q_35/format,webp)
32(cos60^circ +isin60^circ ) Next, find Theta 's for all 5 roots of 16+16sqrt (3)i Theta =60^circ ,420^circ ,[?]^circ ,square ^circ ,square ^circ Remember, to find the different Theta 's add 360^circ to Theta
Solução

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PauloProfissional · Tutor por 6 anos
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To find the roots of the complex number 16 + 16\sqrt{3}i, we first need to express it in polar form.
1. **Convert to Polar Form:**
The magnitude r of the complex number 16 + 16\sqrt{3}i is:
r = \sqrt{16^2 + (16\sqrt{3})^2} = \sqrt{256 + 768} = \sqrt{1024} = 32
The argument \theta is:
\theta = \tan^{-1}\left(\frac{16\sqrt{3}}{16}\right) = \tan^{-1}(\sqrt{3}) = 60^\circ
So, the complex number in polar form is:
32(\cos 60^\circ + i \sin 60^\circ)
2. **Find the Roots:**
The roots of the complex number 32(\cos 60^\circ + i \sin 60^\circ) are given by:
z_k = r^{1/n} \left( \cos \frac{\theta + 360^\circ k}{n} + i \sin \frac{\theta + 360^\circ k}{n} \right)
where n is the number of roots, and k = 0, 1, 2, \ldots, n-1.
For 16 + 16\sqrt{3}i, n = 5 (since it is a 5th degree polynomial).
The roots are:
z_k = 32^{1/5} \left( \cos \frac{60^\circ + 360^\circ k}{5} + i \sin \frac{60^\circ + 360^\circ k}{5} \right)
Simplifying 32^{1/5}:
32^{1/5} = 2
So the roots are:
z_k = 2 \left( \cos \frac{60^\circ + 360^\circ k}{5} + i \sin \frac{60^\circ + 360^\circ k}{5} \right)
3. **Calculate the Specific Roots:**
For k = 0:
z_0 = 2 \left( \cos \frac{60^\circ}{5} + i \sin \frac{60^\circ}{5} \right) = 2 \left( \cos 12^\circ + i \sin 12^\circ \right)
For k = 1:
z_1 = 2 \left( \cos \frac{420^\circ}{5} + i \sin \frac{420^\circ}{5} \right) = 2 \left( \cos 84^\circ + i \sin 84^\circ \right)
For k = 2:
z_2 = 2 \left( \cos \frac{780^\circ}{5} + i \sin \frac{780^\circ}{5} \right) = 2 \left( \cos 156^\circ + i \sin 156^\circ \right)
For k = 3:
z_3 = 2 \left( \cos \frac{1140^\circ}{5} + i \sin \frac{1140^\circ}{5} \right) = 2 \left( \cos 228^\circ + i \sin 228^\circ \right)
For k = 4:
z_4 = 2 \left( \cos \frac{1320^\circ}{5} + i \sin \frac{1320^\circ}{5} \right) = 2 \left( \cos 264^\circ + i \sin 264^\circ \right)
So, the five roots are:
\Theta = 60^\circ, 420^\circ, 12^\circ, 84^\circ, 156^\circ
Adding 360^\circ to each \Theta as needed:
\Theta = 60^\circ, 420^\circ, 372^\circ, 444^\circ, 516^\circ
Thus, the complete list of \Theta values for the roots is:
\[
60^\circ, 420^\circ, 372^\circ, 444^\circ, 516^\circ
1. **Convert to Polar Form:**
The magnitude r of the complex number 16 + 16\sqrt{3}i is:
r = \sqrt{16^2 + (16\sqrt{3})^2} = \sqrt{256 + 768} = \sqrt{1024} = 32
The argument \theta is:
\theta = \tan^{-1}\left(\frac{16\sqrt{3}}{16}\right) = \tan^{-1}(\sqrt{3}) = 60^\circ
So, the complex number in polar form is:
32(\cos 60^\circ + i \sin 60^\circ)
2. **Find the Roots:**
The roots of the complex number 32(\cos 60^\circ + i \sin 60^\circ) are given by:
z_k = r^{1/n} \left( \cos \frac{\theta + 360^\circ k}{n} + i \sin \frac{\theta + 360^\circ k}{n} \right)
where n is the number of roots, and k = 0, 1, 2, \ldots, n-1.
For 16 + 16\sqrt{3}i, n = 5 (since it is a 5th degree polynomial).
The roots are:
z_k = 32^{1/5} \left( \cos \frac{60^\circ + 360^\circ k}{5} + i \sin \frac{60^\circ + 360^\circ k}{5} \right)
Simplifying 32^{1/5}:
32^{1/5} = 2
So the roots are:
z_k = 2 \left( \cos \frac{60^\circ + 360^\circ k}{5} + i \sin \frac{60^\circ + 360^\circ k}{5} \right)
3. **Calculate the Specific Roots:**
For k = 0:
z_0 = 2 \left( \cos \frac{60^\circ}{5} + i \sin \frac{60^\circ}{5} \right) = 2 \left( \cos 12^\circ + i \sin 12^\circ \right)
For k = 1:
z_1 = 2 \left( \cos \frac{420^\circ}{5} + i \sin \frac{420^\circ}{5} \right) = 2 \left( \cos 84^\circ + i \sin 84^\circ \right)
For k = 2:
z_2 = 2 \left( \cos \frac{780^\circ}{5} + i \sin \frac{780^\circ}{5} \right) = 2 \left( \cos 156^\circ + i \sin 156^\circ \right)
For k = 3:
z_3 = 2 \left( \cos \frac{1140^\circ}{5} + i \sin \frac{1140^\circ}{5} \right) = 2 \left( \cos 228^\circ + i \sin 228^\circ \right)
For k = 4:
z_4 = 2 \left( \cos \frac{1320^\circ}{5} + i \sin \frac{1320^\circ}{5} \right) = 2 \left( \cos 264^\circ + i \sin 264^\circ \right)
So, the five roots are:
\Theta = 60^\circ, 420^\circ, 12^\circ, 84^\circ, 156^\circ
Adding 360^\circ to each \Theta as needed:
\Theta = 60^\circ, 420^\circ, 372^\circ, 444^\circ, 516^\circ
Thus, the complete list of \Theta values for the roots is:
\[
60^\circ, 420^\circ, 372^\circ, 444^\circ, 516^\circ
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