IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Find accurate and detailed answers to your questions from our experienced and dedicated community members.

Which expression represents a 5th root of [tex]\(-i\)[/tex]?

A. [tex]\cos \left(\frac{7 \pi}{4}\right) + \sin \left(\frac{7 \pi}{4}\right)[/tex]

B. [tex]\cos \left(\frac{7 \pi}{6}\right) + \sin \left(\frac{7 \pi}{6}\right)[/tex]

C. [tex]\cos \left(\frac{7 \pi}{8}\right) + \sin \left(\frac{7 \pi}{8}\right)[/tex]

D. [tex]\cos \left(\frac{7 \pi}{10}\right) + \sin \left(\frac{7 \pi}{10}\right)[/tex]


Sagot :

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]