To solve for [tex]\(\sin \theta + \cos \theta\)[/tex] given that [tex]\(\sin \theta - \cos \theta = \frac{1}{2}\)[/tex] and [tex]\(\theta\)[/tex] is an acute angle, we can employ some trigonometric identities and algebra.
1. Square both sides of the given equation:
[tex]\[
(\sin \theta - \cos \theta)^2 = \left(\frac{1}{2}\right)^2
\][/tex]
This simplifies to:
[tex]\[
\sin^2 \theta - 2 \sin \theta \cos \theta + \cos^2 \theta = \frac{1}{4}
\][/tex]
2. Use the Pythagorean identity:
Recall that [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]. Substitute this into the equation:
[tex]\[
1 - 2 \sin \theta \cos \theta = \frac{1}{4}
\][/tex]
3. Isolate the [tex]\(-2 \sin \theta \cos \theta\)[/tex] term:
Subtract 1 from both sides:
[tex]\[
-2 \sin \theta \cos \theta = \frac{1}{4} - 1
\][/tex]
Simplify the right-hand side:
[tex]\[
-2 \sin \theta \cos \theta = \frac{1}{4} - \frac{4}{4} = -\frac{3}{4}
\][/tex]
4. Solve for [tex]\(2 \sin \theta \cos \theta\)[/tex]:
Multiply both sides by -1:
[tex]\[
2 \sin \theta \cos \theta = \frac{3}{4}
\][/tex]
5. Recall the double-angle identity:
[tex]\[
2 \sin \theta \cos \theta = \sin 2\theta
\][/tex]
Therefore:
[tex]\[
\sin 2\theta = \frac{3}{4}
\][/tex]
6. Express [tex]\((\sin \theta + \cos \theta)^2\)[/tex]:
We need to find [tex]\(\sin \theta + \cos \theta\)[/tex]. Square this expression:
[tex]\[
(\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta
\][/tex]
Recall from earlier that:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
And we have:
[tex]\[
2 \sin \theta \cos \theta = \frac{3}{4}
\][/tex]
Substitute these into the squared sum expression:
[tex]\[
(\sin \theta + \cos \theta)^2 = 1 + \frac{3}{4} = 1.75
\][/tex]
7. Take the square root of both sides:
[tex]\[
\sin \theta + \cos \theta = \sqrt{1.75} = \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2}
\][/tex]
Therefore, the value of [tex]\(\sin \theta + \cos \theta\)[/tex] is [tex]\(\boxed{\frac{\sqrt{7}}{2}}\)[/tex].
