Question

If cos(O) = 24/25, and is in Quadrant I, then what is cos(0/2)? Simplify your answer completely, rationalize the denominator, and enter it in fractional form.

Answer 1

Explanation:

$\begin{gathered} \cos (\theta)\text{ = }\frac{24}{25} \\ \\ \text{Since it is quadrant 1, then cos }\theta\text{ is positive} \end{gathered}$

Using half angle formula:

$\cos (\frac{\theta}{2})\text{ = }\pm\sqrt[]{\frac{1+\text{ cos}(\theta)}{2}}$$\begin{gathered} \cos (\frac{\theta}{2})\text{ = }\pm\sqrt[]{\frac{1+\frac{24}{25}}{2}} \\ \text{= }\pm\sqrt[]{\frac{\frac{1(25)+24}{25}}{2}}\text{ = }\pm\sqrt[]{\frac{\frac{25+24}{25}}{2}} \\ =\text{ }\pm\sqrt[]{\frac{\frac{49}{25}}{2}} \\ \end{gathered}$$\begin{gathered} =\pm\text{ }\sqrt[]{\frac{49}{25\times2}} \\ =\text{ }\pm\text{ }\frac{7}{5}\text{ }\times\text{ }\sqrt[]{\frac{1}{2}} \\ =\text{ }\pm\text{ }\frac{7}{5}\text{ }\times\text{ }\frac{1}{\sqrt[]{2}} \\ \end{gathered}$

Rationalising the denominator:

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