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Determine the tangent and cotangent of [tex]\(\theta\)[/tex] given the following. Round your answer to three decimal places, if necessary.

[tex]\(\sin (\theta) = -0.5736\)[/tex] and [tex]\(\cos (\theta) = 0.8192\)[/tex]

Answer:
[tex]\[
\begin{array}{l}
\tan (\theta) = \square \\
\cot (\theta) = \square
\end{array}
\][/tex]

Sagot :

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]