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

Step-by-step explanation:

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

Using half angle formula:

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

Rationalising the denominator: