To determine the value of [tex]\(\cot(\alpha)\)[/tex] when [tex]\(\sin(\alpha) = \frac{4}{5}\)[/tex] and [tex]\(\cos(\alpha) = \frac{3}{5}\)[/tex], we use the relationship between the trigonometric functions. The cotangent of an angle [tex]\(\alpha\)[/tex] is given by the ratio of the cosine to the sine of that angle. Mathematically, it is expressed as:
[tex]\[
\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}
\][/tex]
Given the values:
[tex]\[
\sin(\alpha) = \frac{4}{5}
\][/tex]
[tex]\[
\cos(\alpha) = \frac{3}{5}
\][/tex]
We substitute these values into the formula:
[tex]\[
\cot(\alpha) = \frac{\frac{3}{5}}{\frac{4}{5}}
\][/tex]
When dividing fractions, we multiply by the reciprocal of the divisor:
[tex]\[
\cot(\alpha) = \frac{3}{5} \times \frac{5}{4}
\][/tex]
The 5s in the numerator and denominator cancel each other out:
[tex]\[
\cot(\alpha) = \frac{3}{4}
\][/tex]
Hence, the value of [tex]\(\cot(\alpha)\)[/tex] is:
[tex]\[
\cot(\alpha) = 0.75
\][/tex]
Therefore, [tex]\(\cot(\alpha)\)[/tex] is approximately [tex]\(0.75\)[/tex].
