Given that $\cos α = \frac{2}{3}$ and $0 < α < \frac{π}{2}$ , determine the exact value of $\cos \frac{α}{2}$ .
$\cos \frac{α}{2} =$
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)

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Answer

Solving this, we find that the exact value of $\cos \frac{α}{2}$ is $0.9128709291752768$ .

Steps

Step 1 ：We are given that $\cos α = \frac{2}{3}$ and $0 < α < \frac{π}{2}$ . We need to find the value of $\cos \frac{α}{2}$ .

Step 2 ：We can use the half-angle formula for cosine, which is $\cos \frac{α}{2} = \pm \sqrt{\frac{1 + \cos α}{2}}$ .

Step 3 ：Since $0 < α < \frac{π}{2}$ , we know that $0 < \frac{α}{2} < \frac{π}{4}$ , so $\cos \frac{α}{2}$ will be positive.

Step 4 ：Therefore, we can ignore the negative root and use the formula $\cos \frac{α}{2} = \sqrt{\frac{1 + \cos α}{2}}$ to find the value of $\cos \frac{α}{2}$ .

Step 5 ：Substituting the given value of $\cos α$ into the formula, we get $\cos \frac{α}{2} = \sqrt{\frac{1 + \frac{2}{3}}{2}}$ .

Step 6 ：Solving this, we find that the exact value of $\cos \frac{α}{2}$ is $0.9128709291752768$ .

Given that cos⁡α=23 and 0<α<π2, determine the exact value of cos⁡α2.cos⁡α2=(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)

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Given that $\cos α = \frac{2}{3}$ and $0 < α < \frac{π}{2}$ , determine the exact value of $\cos \frac{α}{2}$ .
$\cos \frac{α}{2} =$
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)