To determine the tangent and cotangent of [tex]\(\theta\)[/tex], given that [tex]\(\sin(\theta) = -0.5736\)[/tex] and [tex]\(\cos(\theta) = 0.8192\)[/tex], we follow these steps:
1. Calculate the Tangent of [tex]\(\theta\)[/tex]:
The tangent of an angle is defined as the ratio of the sine to the cosine of that angle. Mathematically, this is written as:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
Substituting the provided values:
[tex]\[
\tan(\theta) = \frac{-0.5736}{0.8192}
\][/tex]
Performing the division gives:
[tex]\[
\tan(\theta) \approx -0.7
\][/tex]
2. Calculate the Cotangent of [tex]\(\theta\)[/tex]:
The cotangent of an angle is defined as the reciprocal of the tangent of that angle. Mathematically, this is written as:
[tex]\[
\cot(\theta) = \frac{1}{\tan(\theta)}
\][/tex]
Using the previously calculated value of [tex]\(\tan(\theta)\)[/tex]:
[tex]\[
\cot(\theta) = \frac{1}{-0.7}
\][/tex]
Performing the division gives:
[tex]\[
\cot(\theta) \approx -1.428
\][/tex]
Therefore, the values of the tangent and cotangent of [tex]\(\theta\)[/tex], rounded to three decimal places, are:
[tex]\[
\begin{array}{l}
\tan(\theta) = -0.7 \\
\cot(\theta) = -1.428
\end{array}
\][/tex]
