use an identity to write the expression as a single trigonometric function. \n\n\\( \\sqrt { \\frac { 1 +…

use an identity to write the expression as a single trigonometric function. \n\n\\( \\sqrt { \\frac { 1 + \\cos 82 ^ { \\circ } } { 2 } } \\)\n\n\\( \\sqrt { \\frac { 1 + \\cos 82 ^ { \\circ } } { 2 } } = \\square ^ { \\circ } \\) (type an exact answer.)

use an identity to write the expression as a single trigonometric function. \n\n\\( \\sqrt { \\frac { 1 + \\cos 82 ^ { \\circ } } { 2 } } \\)\n\n\\( \\sqrt { \\frac { 1 + \\cos 82 ^ { \\circ } } { 2 } } = \\square ^ { \\circ } \\) (type an exact answer.)

Answer

Explanation:

Step1: Recall the half - angle identity

The half - angle identity for cosine is (\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}).

Step2: Identify (\alpha)

In the given expression (\sqrt{\frac{1+\cos82^{\circ}}{2}}), we have (\alpha = 82^{\circ}).

Step3: Apply the half - angle identity

Since (82^{\circ}) is in the first quadrant (where cosine is positive), and using the half - angle identity (\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}) with (\alpha = 82^{\circ}), we get (\cos\frac{82^{\circ}}{2}).

Step4: Calculate (\frac{\alpha}{2})

(\frac{82^{\circ}}{2}=41^{\circ})

Answer:

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