To find the value of [tex]\(\tan \theta\)[/tex] given that [tex]\(\sin \theta = \frac{4}{5}\)[/tex] and [tex]\(\theta\)[/tex] is in quadrant 2, we can follow these steps:
1. Identify the Given Information and Quadrant:
- [tex]\(\sin \theta = \frac{4}{5}\)[/tex]
- [tex]\(\theta\)[/tex] is in quadrant 2.
2. Recall the Sign Conventions in Quadrant 2:
- In quadrant 2, sine is positive and cosine is negative.
3. Use the Pythagorean Identity:
- The Pythagorean identity states that [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex].
- Substitute [tex]\(\sin \theta = \frac{4}{5}\)[/tex] into the identity:
[tex]\[
\left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1
\][/tex]
- Simplify the equation:
[tex]\[
\frac{16}{25} + \cos^2 \theta = 1
\][/tex]
- Solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}
\][/tex]
- Take the square root of both sides to find [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5}
\][/tex]
- Since [tex]\(\theta\)[/tex] is in quadrant 2, where cosine is negative:
[tex]\[
\cos \theta = -\frac{3}{5}
\][/tex]
4. Calculate Tangent:
- Tangent is defined as [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex].
- Substitute the known values of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{5} \div -\frac{3}{5} = \frac{4}{5} \times -\frac{5}{3} = \frac{4 \times -5}{5 \times 3} = -\frac{20}{15} = -\frac{4}{3}
\][/tex]
Therefore, the value of [tex]\(\tan \theta\)[/tex] is:
[tex]\[
\tan \theta = -\frac{4}{3}
\][/tex]
