If $\cos \theta=\frac{2}{3}$, then what is the positive value of $\sin \frac{1}{2} \theta$, in simplest radical form with a rational denominator?
Answer
Using the half-angle formula for sine: \(\sin \frac{1}{2} \theta = \pm \sqrt{\frac{1 - \cos \theta}{2}}\), we can plug in the value of \(\cos \theta\) and simplify the expression: \(\sin \frac{1}{2} \theta = \sqrt{\frac{1 - \frac{2}{3}}{2}} = \boxed{\frac{\sqrt{6}}{6}}\)
Step 1 ：Given that \(\cos \theta = \frac{2}{3}\), we need to find the positive value of \(\sin \frac{1}{2} \theta\).
Step 2 ：Using the half-angle formula for sine: \(\sin \frac{1}{2} \theta = \pm \sqrt{\frac{1 - \cos \theta}{2}}\), we can plug in the value of \(\cos \theta\) and simplify the expression: \(\sin \frac{1}{2} \theta = \sqrt{\frac{1 - \frac{2}{3}}{2}} = \boxed{\frac{\sqrt{6}}{6}}\)
