Question

i need help with number 22.i have the formula cos theta/2 = plus or minus the square root of (1 + cos theta)/2 which i changed to cos beta/2 = plus or minus the square root of (1 + cos beta)/2.i put in -12/13 for cos beta and got: plus or minus the square root of (1/13)/2. i then did keep change flip and multiplied 1/13 • 1/2 and got 1/26. i’m stuck here and don’t know how to do the square root of 1/26

Answer 1

$cos\beta=-\frac{12}{13}$

Use the next identity to find cosβ/2:

$cos(\frac{\theta}{2})=\pm\sqrt{\frac{1+cos\theta}{2}}$$\begin{gathered} cos(\frac{\beta}{2})=\pm\sqrt{\frac{1+(-\frac{12}{13})}{2}}=\pm\sqrt{\frac{1-\frac{12}{13}}{2}}=\pm\sqrt{\frac{\frac{13-12}{13}}{2}}=\pm\sqrt{\frac{\frac{1}{13}}{2}}=\pm\sqrt{\frac{1}{26}}=\pm\frac{\sqrt{1}}{\sqrt{26}}=\pm\frac{1}{\sqrt{26}} \\ \\ Rationalize\text{ the denominator:} \\ \pm\frac{1}{\sqrt{26}}*\frac{\sqrt{26}}{\sqrt{26}}=\pm\frac{\sqrt{26}}{26} \end{gathered}$

Then, the cos(β/2) is ± √26 /26