To solve the given problem, let's analyze the trigonometric identities and the information provided:
Given:
- [tex]\(\tan \theta = \frac{3}{4}\)[/tex]
- [tex]\( \theta \)[/tex] is in the third quadrant.
We know the following about the third quadrant:
- In the third quadrant, both sine ([tex]\(\sin \theta\)[/tex]) and cosine ([tex]\(\cos \theta\)[/tex]) are negative.
Now, let's delve into the trigonometric identities and relationships:
1. [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex]
Given that [tex]\(\tan \theta = \frac{3}{4}\)[/tex], we can consider a corresponding right triangle where the opposite side is 3 and the adjacent side is 4. Now, we need to find the hypotenuse using the Pythagorean theorem:
[tex]\[ \text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]
With the hypotenuse calculated as 5, we can now find [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \][/tex]
[tex]\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \][/tex]
Considering that [tex]\(\theta\)[/tex] is in the third quadrant, where both sine and cosine are negative:
[tex]\[ \sin \theta = -\frac{3}{5} \][/tex]
[tex]\[ \cos \theta = -\frac{4}{5} \][/tex]
Next, let's calculate [tex]\(\csc \theta\)[/tex] and [tex]\(\cot \theta\)[/tex]:
[tex]\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} \][/tex]
[tex]\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \][/tex]
Finally, let's address each option provided:
A. [tex]\(\sin \theta = \frac{3}{5}\)[/tex] — Incorrect (it should be [tex]\(-\frac{3}{5}\)[/tex])
B. [tex]\(\cos \theta = -\frac{4}{5}\)[/tex] — Correct
C. [tex]\(\csc \theta = -\frac{5}{3}\)[/tex] — Correct
D. [tex]\(\cot \theta = \frac{4}{3}\)[/tex] — Correct
Thus, the correct details for [tex]\(\sin \theta\)[/tex], [tex]\(\cos \theta\)[/tex], [tex]\(\csc \theta\)[/tex], and [tex]\(\cot \theta\)[/tex] are:
[tex]\[ \sin \theta = -\frac{3}{5} \][/tex]
[tex]\[ \cos \theta = -\frac{4}{5} \][/tex]
[tex]\[ \csc \theta = -\frac{5}{3} \][/tex]
[tex]\[ \cot \theta = \frac{4}{3} \][/tex]
From the problem, the correct options are B, C, and D.
