what is the exact value of \\( \\cos ( \\frac { 7 \\pi } { 8 } ) \\)?\n\n\\( - \\frac { \\sqrt { 2 - \\sqrt…

what is the exact value of \\( \\cos ( \\frac { 7 \\pi } { 8 } ) \\)?\n\n\\( - \\frac { \\sqrt { 2 - \\sqrt { 2 } } } { 4 } \\)\n\n\\( - \\frac { \\sqrt { 2 + \\sqrt { 2 } } } { 4 } \\)\n\n\\( - \\frac { \\sqrt { 2 - \\sqrt { 2 } } } { 2 } \\)\n\n\\( - \\frac { \\sqrt { 2 + \\sqrt { 2 } } } { 2 } \\)

what is the exact value of \\( \\cos ( \\frac { 7 \\pi } { 8 } ) \\)?\n\n\\( - \\frac { \\sqrt { 2 - \\sqrt { 2 } } } { 4 } \\)\n\n\\( - \\frac { \\sqrt { 2 + \\sqrt { 2 } } } { 4 } \\)\n\n\\( - \\frac { \\sqrt { 2 - \\sqrt { 2 } } } { 2 } \\)\n\n\\( - \\frac { \\sqrt { 2 + \\sqrt { 2 } } } { 2 } \\)

Answer

Answer:

D. (-\frac{\sqrt{2 + \sqrt{2}}}{2})

Explanation:

Step1: Use the half - angle formula

The half - angle formula for cosine is (\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}). We know that (\frac{7\pi}{8}=\frac{\frac{7\pi}{4}}{2}), and (\cos\frac{7\pi}{8}<0) (since (\frac{7\pi}{8}) is in the second quadrant where cosine is negative).

Step2: Substitute (\alpha=\frac{7\pi}{4}) into the formula

We know that (\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}). Substitute into (\cos\frac{\alpha}{2}=-\sqrt{\frac{1+\cos\alpha}{2}}) (negative because of the quadrant), we get (\cos\frac{7\pi}{8}=-\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}).

Step3: Simplify the expression

[ \begin{align*} -\sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}&=-\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}\ &=-\sqrt{\frac{2+\sqrt{2}}{4}}\ &=-\frac{\sqrt{2+\sqrt{2}}}{2} \end{align*} ]