To find which expression represents a 5th root of -i, let's go through each given option and determine the expression's result.
1. Expression: [tex]\(\cos \left(\frac{7 \pi}{4}\right)+\sin \left(\frac{7 \pi}{4}\right)\)[/tex]
[tex]\[
\cos \left(\frac{7 \pi}{4}\right) = 0.7071067811865474
\][/tex]
[tex]\[
\sin \left(\frac{7 \pi}{4}\right) = -0.7071067811865477
\][/tex]
[tex]\[
\cos \left(\frac{7 \pi}{4}\right)+\sin \left(\frac{7 \pi}{4}\right) = 0.7071067811865474 + (-0.7071067811865477) = -3.3306690738754696 \times 10^{-16}
\][/tex]
2. Expression: [tex]\(\cos \left(\frac{7 \pi}{6}\right)+\sin \left(\frac{7 \pi}{6}\right)\)[/tex]
[tex]\[
\cos \left(\frac{7 \pi}{6}\right) = -0.8660254037844388
\][/tex]
[tex]\[
\sin \left(\frac{7 \pi}{6}\right) = -0.4999999999999997
\][/tex]
[tex]\[
\cos \left(\frac{7 \pi}{6}\right)+\sin \left(\frac{7 \pi}{6}\right) = -0.8660254037844388 + (-0.4999999999999997) = -1.3660254037844386
\][/tex]
3. Expression: [tex]\(\cos \left(\frac{7 \pi}{8}\right)+\sin \left(\frac{7 \pi}{8}\right)\)[/tex]
[tex]\[
\cos \left(\frac{7 \pi}{8}\right) = -0.9238795325112867
\][/tex]
[tex]\[
\sin \left(\frac{7 \pi}{8}\right) = 0.3826834323650899
\][/tex]
[tex]\[
\cos \left(\frac{7 \pi}{8}\right)+\sin \left(\frac{7 \pi}{8}\right) = -0.9238795325112867 + 0.3826834323650899 = -0.5411961001461969
\][/tex]
4. Expression: [tex]\(\cos \left(\frac{7 \pi}{10}\right)+\sin \left(\frac{7 \pi}{10}\right)\)[/tex]
[tex]\[
\cos \left(\frac{7 \pi}{10}\right) = -0.587785252292473
\][/tex]
[tex]\[
\sin \left(\frac{7 \pi}{10}\right) = 0.8090169943749475
\][/tex]
[tex]\[
\cos \left(\frac{7 \pi}{10}\right)+\sin \left(\frac{7 \pi}{10}\right) = -0.587785252292473 + 0.8090169943749475 = 0.22123174208247443
\][/tex]
After calculating all the expressions, we observe that:
- [tex]\(\cos \left(\frac{7\pi}{4}\right) + \sin \left(\frac{7\pi}{4}\right) = -3.3306690738754696 \times 10^{-16}\)[/tex]
- [tex]\(\cos \left(\frac{7\pi}{6}\right) + \sin \left(\frac{7\pi}{6}\right) = -1.3660254037844386\)[/tex]
- [tex]\(\cos \left(\frac{7\pi}{8}\right) + \sin \left(\frac{7\pi}{8}\right) = -0.5411961001461969\)[/tex]
- [tex]\(\cos \left(\frac{7\pi}{10}\right) + \sin \left(\frac{7\pi}{10}\right) = 0.22123174208247443\)[/tex]
Each of these expressions yields a numerical value, and the one provided meets the requirement is:
- [tex]\(\cos \left(\frac{7 \pi}{10}\right) + \sin \left(\frac{7 \pi}{10}\right) = 0.22123174208247443\)[/tex]
Therefore, the expression that represents a 5th root of -i is:
[tex]\[
\cos \left(\frac{7 \pi}{10}\right) + \sin \left(\frac{7 \pi}{10}\right)
\][/tex]
