To solve the problem where [tex]\(\tan \theta = \sqrt{3}\)[/tex] and [tex]\(\theta\)[/tex] is an acute angle, we need to find the value of [tex]\(1 + \cos \theta\)[/tex]. Here is the detailed step-by-step solution:
1. Identify the angle [tex]\(\theta\)[/tex]:
Since [tex]\(\tan \theta = \sqrt{3}\)[/tex] and [tex]\(\theta\)[/tex] is an acute angle, we can recognize that [tex]\(\theta = 60^\circ\)[/tex] because [tex]\(\tan 60^\circ = \sqrt{3}\)[/tex].
2. Determine [tex]\(\cos \theta\)[/tex]:
Once we know [tex]\(\theta = 60^\circ\)[/tex], we can find [tex]\(\cos \theta\)[/tex]. From trigonometric ratios, we know that:
[tex]\[
\cos 60^\circ = \frac{1}{2}
\][/tex]
3. Calculate [tex]\(1 + \cos \theta\)[/tex]:
Substitute the value of [tex]\(\cos \theta\)[/tex]:
[tex]\[
1 + \cos 60^\circ = 1 + \frac{1}{2}
\][/tex]
4. Simplify the expression:
[tex]\[
1 + \frac{1}{2} = 1 + 0.5 = 1.5
\][/tex]
Therefore, the value of [tex]\(1 + \cos \theta\)[/tex] is [tex]\(1.5\)[/tex].
