To determine [tex]\(\cos P\)[/tex] in the right triangle [tex]\(\triangle ZAP\)[/tex] with right angle [tex]\(A\)[/tex] and [tex]\(\sin Z = \frac{4}{5}\)[/tex], we can use the properties of right triangles and trigonometric identities.
1. Recognize the relationship involving sine:
[tex]\(\sin Z = \frac{\text{Opposite side to } Z}{\text{Hypotenuse}}\)[/tex]. Given [tex]\(\sin Z = \frac{4}{5}\)[/tex], it means in [tex]\(\triangle ZAP\)[/tex], if we denote the sides relative to angle [tex]\(Z\)[/tex]:
- Opposite to [tex]\(Z\)[/tex] = 4
- Hypotenuse = 5
2. Find the adjacent side to [tex]\(Z\)[/tex] using the Pythagorean theorem:
[tex]\[
\text{Hypotenuse}^2 = (\text{Opposite side})^2 + (\text{Adjacent side})^2
\][/tex]
Substituting the known values:
[tex]\[
5^2 = 4^2 + (\text{Adjacent side})^2
\][/tex]
[tex]\[
25 = 16 + (\text{Adjacent side})^2
\][/tex]
[tex]\[
(\text{Adjacent side})^2 = 25 - 16 = 9
\][/tex]
[tex]\[
\text{Adjacent side} = \sqrt{9} = 3
\][/tex]
3. Calculate [tex]\(\cos Z\)[/tex]:
[tex]\[
\cos Z = \frac{\text{Adjacent side to } Z}{\text{Hypotenuse}} = \frac{3}{5}
\][/tex]
4. Use the fact that in a right triangle, the sine of one acute angle is the cosine of the other:
Since [tex]\(\triangle ZAP\)[/tex] is a right triangle and [tex]\(A\)[/tex] is the right angle:
[tex]\[
P = 90^\circ - Z
\][/tex]
Hence, [tex]\(\cos P = \sin Z\)[/tex].
Given [tex]\(\sin Z = \frac{4}{5}\)[/tex], we have:
[tex]\[
\cos P = \frac{4}{5}
\][/tex]
Therefore, the value of [tex]\(\cos P\)[/tex] is [tex]\(\frac{4}{5}\)[/tex].
The correct answer is:
[tex]\[
\boxed{\frac{4}{5}}
\][/tex]
