Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

The point [tex]\(\left(\frac{5}{13}, y\right)\)[/tex] in the fourth quadrant corresponds to angle [tex]\(\theta\)[/tex] on the unit circle.

[tex]\[
\begin{array}{l}
\sec \theta = \square \\
\cot \theta = \square
\end{array}
\][/tex]



Answer :

Sure, let's break down the problem step by step.

Given:
- The point [tex]\(\left(\frac{5}{13}, y\right)\)[/tex] is in the fourth quadrant on the unit circle.

First, we know that the coordinates of any point [tex]\((x, y)\)[/tex] on the unit circle satisfy the equation [tex]\(x^2 + y^2 = 1\)[/tex].

1. Identify [tex]\( x = \frac{5}{13} \)[/tex].

Next, solve for [tex]\( y \)[/tex] using the unit circle equation:

[tex]\[ \left( \frac{5}{13} \right)^2 + y^2 = 1 \][/tex]

[tex]\[ \left( \frac{25}{169} \right) + y^2 = 1 \][/tex]

[tex]\[ y^2 = 1 - \frac{25}{169} \][/tex]

[tex]\[ y^2 = \frac{169}{169} - \frac{25}{169} \][/tex]

[tex]\[ y^2 = \frac{144}{169} \][/tex]

[tex]\[ y = \pm \frac{\sqrt{144}}{\sqrt{169}} \][/tex]

[tex]\[ y = \pm \frac{12}{13} \][/tex]

Since the point is in the fourth quadrant, [tex]\(y\)[/tex] is negative:

[tex]\[ y = -\frac{12}{13} \][/tex]

Now we need to find [tex]\(\sec \theta\)[/tex] and [tex]\(\cot \theta\)[/tex]:

2. Compute [tex]\(\sec \theta\)[/tex]:

[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{x} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \][/tex]

Therefore, [tex]\(\sec \theta = 2.6\)[/tex].

3. Compute [tex]\(\cot \theta\)[/tex]:

[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{x}{y} = \frac{\frac{5}{13}}{-\frac{12}{13}} = \frac{5}{-12} = - \frac{5}{12} \][/tex]

Therefore, [tex]\(\cot \theta = -0.4166666666666667\)[/tex].

So the answers are:

[tex]\[ \sec \theta = 2.6 \][/tex]

[tex]\[ \cot \theta = -0.4166666666666667 \][/tex]