Certainly! Let's break this down step by step.
1. Understand the Given Information:
We are given that [tex]\(\sin A = \frac{4}{5}\)[/tex].
2. Relate [tex]\(\sin A\)[/tex] to a Right Triangle:
In a right triangle, [tex]\(\sin A\)[/tex] (sine of angle [tex]\(A\)[/tex]) is the ratio of the length of the opposite side to the hypotenuse. Therefore, if [tex]\(\sin A = \frac{4}{5}\)[/tex], we can consider:
- Opposite side = 4
- Hypotenuse = 5
3. Use the Pythagorean Theorem to Find the Adjacent Side:
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The relationship is:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Substituting the values, we get:
[tex]\[
\text{adjacent}^2 + 4^2 = 5^2
\][/tex]
4. Solve for the Adjacent Side:
[tex]\[
\text{adjacent}^2 + 16 = 25
\][/tex]
[tex]\[
\text{adjacent}^2 = 25 - 16
\][/tex]
[tex]\[
\text{adjacent}^2 = 9
\][/tex]
[tex]\[
\text{adjacent} = \sqrt{9} = 3
\][/tex]
5. Calculate [tex]\(\tan A\)[/tex]:
The tangent of angle [tex]\(A\)[/tex] ([tex]\(\tan A\)[/tex]) is the ratio of the length of the opposite side to the length of the adjacent side. Therefore:
[tex]\[
\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}
\][/tex]
6. Conclusion:
The adjacent side is 3 and [tex]\(\tan A = \frac{4}{3}\)[/tex].
So, the value of [tex]\(\tan A\)[/tex] is [tex]\(\frac{4}{3}\)[/tex] or approximately 1.3333333333333333.
