Use the half-angle formula for cosine to compute $\cos(\theta/2)$ given $\cos(\theta)=63/68$ where $0\lt\theta\lt\pi/2$ Use the half-angle formula for cosine to compute $\cos(\theta/2)$ given $\cos(\theta)=63/68$ where $0\lt\theta\lt\pi/2$.

I know that $\cos(\theta/2)= \pm\sqrt{\frac{\cos(\theta)+1}{2}}$.  Therefore the answer would be $\pm\sqrt{\frac{(63/68)+1}{2}}$.  However, I don't know whether or not its negative or positive.
Given that $0 &lt; \theta &lt; \pi/2$, it follows that the cosine function is positive, allowing you to determine a sign. Always go back to the graph when in doubt.
T. Bongers is right: When an angle is on the interval  $0 &lt; \theta &lt; 2\pi$, the cosine of the angle is positive. More generally, the six basic circular trig functions { $\sin(x)$, $\cos(x)$, $\tan(x)$, $\csc(x)$, $\sec(x)$, $\cot(x)$ } follow these patterns:


Quadrant I: All six functions are positive.
Quadrant II: $\sin(x)$ and $\csc(x)$ are positive.
Quadrant III: $\tan(x)$ and $\cot(x)$ are positive.
Quadrant IV: $\cos(x)$ and $\sec(x)$ are positive.


The way my Calc teacher tells me to remember this pattern (or maybe it's called a mnemonic?) is cycling through the Quadrants in order: "All Students Take Calculus!" The first letter of each word corresponds to the basic function(s) that produces positive values in that Quadrant. A: all functions; S: $\sin(x)$; T: $\tan(x)$; C: $\cos(x)$. Because the inverse of a number does not change its sign, the reciprocal functions of those mentioned above follow the same positive negative patterns as the 3 functions I just mentioned. With any Quadrantal angle (when $x$ is a multiple of 0, 90, 180, or 270 degrees), each of the aforementioned functions has its own set of special cases that are more easily understood by practicing to make the graphs of the functions.