3. (6 pts) If [tex]\csc (\theta) = 2[/tex] and [tex]\theta[/tex] is in Quadrant I, find the values of the six trig functions of [tex]\theta[/tex]. State "undefined" if a function is undefined.

\begin{tabular}{|l|l|l|}
\hline
[tex]$\sin (\theta)=$[/tex] & [tex]$\cos (\theta)=$[/tex] & [tex]$\tan (\theta)=$[/tex] \\
\hline
[tex]$\csc (\theta)=$[/tex] & [tex]$\sec (\theta)=$[/tex] & [tex]$\cot (\theta)=$[/tex] \\
\hline
\end{tabular}



Answer :

Given that [tex]\(\csc(\theta) = 2\)[/tex] and [tex]\(\theta\)[/tex] is in Quadrant I, we need to find the values of the six trigonometric functions of [tex]\(\theta\)[/tex]. Here are the step-by-step calculations:

1. Finding [tex]\(\sin(\theta)\)[/tex]:

[tex]\(\csc(\theta)\)[/tex] is the reciprocal of [tex]\(\sin(\theta)\)[/tex]. Therefore:
[tex]\[ \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{2} = 0.5 \][/tex]

2. Finding [tex]\(\cos(\theta)\)[/tex]:

We use the Pythagorean identity [tex]\(\sin^2(\theta) + \cos^2(\theta) = 1\)[/tex]. Given [tex]\(\sin(\theta) = 0.5\)[/tex]:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - (0.5)^2 = 1 - 0.25 = 0.75 \][/tex]
Taking the positive square root (since [tex]\(\theta\)[/tex] is in Quadrant I where [tex]\(\cos(\theta)\)[/tex] is positive):
[tex]\[ \cos(\theta) = \sqrt{0.75} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \approx 0.8660254037844386 \][/tex]

3. Finding [tex]\(\tan(\theta)\)[/tex]:

[tex]\(\tan(\theta)\)[/tex] is the ratio of [tex]\(\sin(\theta)\)[/tex] to [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{0.5}{0.8660254037844386} \approx 0.5773502691896258 \][/tex]

4. Finding [tex]\(\csc(\theta)\)[/tex]:

This was given directly in the problem:
[tex]\[ \csc(\theta) = 2 \][/tex]

5. Finding [tex]\(\sec(\theta)\)[/tex]:

[tex]\(\sec(\theta)\)[/tex] is the reciprocal of [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{0.8660254037844386} \approx 1.1547005383792517 \][/tex]

6. Finding [tex]\(\cot(\theta)\)[/tex]:

[tex]\(\cot(\theta)\)[/tex] is the reciprocal of [tex]\(\tan(\theta)\)[/tex]:
[tex]\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{0.5773502691896258} \approx 1.732050807568877 \][/tex]

Summarizing the results, we get:

[tex]\[ \begin{array}{|l|l|l|} \hline \sin(\theta) = 0.5 & \cos(\theta) = 0.8660254037844386 & \tan(\theta) = 0.5773502691896258 \\ \hline \csc(\theta) = 2 & \sec(\theta) = 1.1547005383792517 & \cot(\theta) = 1.732050807568877 \\ \hline \end{array} \][/tex]