To find [tex]\(\sin \theta\)[/tex] given that [tex]\(0^{\circ} < \theta < 90^{\circ}\)[/tex] and [tex]\(\tan \theta = \frac{4}{3}\)[/tex], we can use trigonometric identities and the Pythagorean theorem.
Here's the step-by-step solution:
1. Understanding [tex]\(\tan \theta\)[/tex]:
The tangent of an angle [tex]\(\theta\)[/tex] is the ratio of the opposite side to the adjacent side in a right-angled triangle. Given [tex]\(\tan \theta = \frac{4}{3}\)[/tex], we can interpret this as:
[tex]\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}
\][/tex]
Therefore, let the opposite side be 4 units and the adjacent side be 3 units.
2. Finding the hypotenuse:
Use the Pythagorean theorem to find the hypotenuse of the triangle. The Pythagorean theorem states:
[tex]\[
\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2
\][/tex]
Here, the hypotenuse [tex]\(h\)[/tex] is calculated as:
[tex]\[
h = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\][/tex]
3. Finding [tex]\(\sin \theta\)[/tex]:
The sine of an angle [tex]\(\theta\)[/tex] in a right-angled triangle is the ratio of the length of the opposite side to the hypotenuse:
[tex]\[
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5}
\][/tex]
4. Conclusion:
Based on the given options, the correct value for [tex]\(\sin \theta\)[/tex] is:
[tex]\[
\frac{4}{5}
\][/tex]
Thus, [tex]\(\sin \theta\)[/tex] is [tex]\(\boxed{\frac{4}{5}}\)[/tex].
