2. If [tex]$0^{\circ}\ \textless \ \theta\ \textless \ 90^{\circ}$[/tex] and [tex]$\tan \theta=\frac{4}{3}$[/tex], find [tex][tex]$\sin \theta$[/tex][/tex] using trigonometric identities.

A. [tex]\frac{3}{5}[/tex]
B. [tex]\frac{3}{4}[/tex]
C. [tex]\frac{1}{4}[/tex]
D. [tex]\frac{4}{5}[/tex]



Answer :

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].