simplify the expression.\n\n\\( \\sqrt { \\frac { 1 - \\cos 157 ^ { \\circ } } { 1 + \\cos 157 ^ { \\circ }…

simplify the expression.\n\n\\( \\sqrt { \\frac { 1 - \\cos 157 ^ { \\circ } } { 1 + \\cos 157 ^ { \\circ } } } \\)\n\n\\( \\sqrt { \\frac { 1 - \\cos 157 ^ { \\circ } } { 1 + \\cos 157 ^ { \\circ } } } = \\square ^ { \\circ } \\)\n(simplify your answer. use integers or decimals for any numbers in the expression. do not include the degree symbol in your answer.)

simplify the expression.\n\n\\( \\sqrt { \\frac { 1 - \\cos 157 ^ { \\circ } } { 1 + \\cos 157 ^ { \\circ } } } \\)\n\n\\( \\sqrt { \\frac { 1 - \\cos 157 ^ { \\circ } } { 1 + \\cos 157 ^ { \\circ } } } = \\square ^ { \\circ } \\)\n(simplify your answer. use integers or decimals for any numbers in the expression. do not include the degree symbol in your answer.)

Answer

Explanation:

Step1: Use the trigonometric identity

We know the identity (\tan\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{1+\cos\alpha}}). Here (\alpha = 157^{\circ}), and (\frac{\alpha}{2}=\frac{157^{\circ}}{2}=78.5^{\circ}). Since (0^{\circ}<78.5^{\circ}<90^{\circ}), (\tan\frac{157^{\circ}}{2}>0).

Step2: Simplify the expression

So (\sqrt{\frac{1-\cos157^{\circ}}{1 + \cos157^{\circ}}}=\tan\frac{157^{\circ}}{2}). Using a calculator, (\tan78.5^{\circ}\approx5) (rounded to the nearest integer).

Answer:

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