Question

If $\cos x = \frac{24}{25}$ and $x$ lies in first quadrant, then $\sin \frac{x}{2} + \cos \frac{x}{2} =$

• A. $\frac{6}{5\sqrt{2}}$
• B. $\frac{8}{5\sqrt{2}}$
• C. $\frac{7}{5\sqrt{2}}$
• D. $\frac{1}{5\sqrt{2}}$

Hint:

An alternative shortcut is to square the expression: $\left(\sin \frac{x}{2} + \cos \frac{x}{2}\right)^2 = \sin^2 \frac{x}{2} + \cos^2 \frac{x}{2} + 2\sin \frac{x}{2}\cos \frac{x}{2} = 1 + \sin x$. Since $\cos x = 24/25$, $\sin x = 7/25$. $1 + 7/25 = 32/25$. Taking the square root gives $\sqrt{32}/5 = 4\sqrt{2}/5 = 8/(5\sqrt{2})$.

Answer 1

The Correct Option is B

Step 1: Note what is given.
We know $\cos x=\frac{24}{25}$ and $x$ is in the first quadrant. We must find $\sin\frac{x}{2}+\cos\frac{x}{2}$.

Step 2: Choose half-angle formulas.
The handy formulas are \[ \sin\frac{x}{2}=\sqrt{\frac{1-\cos x}{2}},\qquad \cos\frac{x}{2}=\sqrt{\frac{1+\cos x}{2}}. \]

Step 3: Decide the signs.
Since $x$ is in the first quadrant, $\frac{x}{2}$ is also in the first quadrant, so both half-angle values are positive. We can safely take the positive roots.

Step 4: Compute $\sin\frac{x}{2}$.
\[ \sin\frac{x}{2}=\sqrt{\frac{1-\frac{24}{25}}{2}}=\sqrt{\frac{\frac{1}{25}}{2}}=\sqrt{\frac{1}{50}}=\frac{1}{5\sqrt2}. \]

Step 5: Compute $\cos\frac{x}{2}$.
\[ \cos\frac{x}{2}=\sqrt{\frac{1+\frac{24}{25}}{2}}=\sqrt{\frac{\frac{49}{25}}{2}}=\sqrt{\frac{49}{50}}=\frac{7}{5\sqrt2}. \]

Step 6: Add them.
They share the same bottom, so \[ \frac{1}{5\sqrt2}+\frac{7}{5\sqrt2}=\frac{8}{5\sqrt2}. \]

Step 7: State the answer.
The sum is $\frac{8}{5\sqrt2}$, which is option (2).
\[ \boxed{\dfrac{8}{5\sqrt2}} \]