Let's solve the problem step-by-step using the given information: [tex]\(\sec \theta = \frac{5}{3}\)[/tex] and the terminal point is in the 4th quadrant.
1. Calculate [tex]\(\cos \theta\)[/tex] using [tex]\(\sec \theta\)[/tex]:
[tex]\(\sec \theta\)[/tex] is the reciprocal of [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta = \frac{1}{\sec \theta} = \frac{1}{\frac{5}{3}} = \frac{3}{5}
\][/tex]
Since we are in the 4th quadrant, [tex]\(\cos \theta\)[/tex] is positive.
2. Calculate [tex]\(\sin \theta\)[/tex] using the Pythagorean identity:
The Pythagorean identity states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substituting [tex]\(\cos \theta = \frac{3}{5}\)[/tex]:
[tex]\[
\sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1
\][/tex]
[tex]\[
\sin^2 \theta + \frac{9}{25} = 1
\][/tex]
[tex]\[
\sin^2 \theta = 1 - \frac{9}{25}
\][/tex]
[tex]\[
\sin^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
\][/tex]
[tex]\[
\sin \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}
\][/tex]
Since we are in the 4th quadrant, [tex]\(\sin \theta\)[/tex] is negative:
[tex]\[
\sin \theta = -\frac{4}{5}
\][/tex]
3. Calculate [tex]\(\csc \theta\)[/tex]:
[tex]\(\csc \theta\)[/tex] is the reciprocal of [tex]\(\sin \theta\)[/tex]:
[tex]\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}
\][/tex]
4. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\(\tan \theta\)[/tex] is the ratio of [tex]\(\sin \theta\)[/tex] to [tex]\(\cos \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{4}{5}}{\frac{3}{5}} = -\frac{4}{3}
\][/tex]
Now we can match these values with the given options:
- [tex]\(\tan \theta = -\frac{4}{3}\)[/tex], which does not match option A.
- [tex]\(\csc \theta = -\frac{5}{4}\)[/tex], which matches option B.
- [tex]\(\sin \theta = -\frac{4}{5}\)[/tex], which does not match option C.
- [tex]\(\cos \theta = \frac{3}{5}\)[/tex], which matches option D.
Therefore, the correct options are:
B. [tex]\(\csc \theta = -\frac{5}{4}\)[/tex]
D. [tex]\(\cos \theta = \frac{3}{5}\)[/tex]
