To solve the problem given that [tex]\(\cot{\theta} = \frac{3}{4}\)[/tex] and the angle [tex]\(\theta\)[/tex] is in the third quadrant, let’s proceed step-by-step:
1. Identify relevant trigonometric relationships:
[tex]\(\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} = \frac{3}{4}\)[/tex]
2. Analyze the quadrant information:
Since [tex]\(\theta\)[/tex] is in the third quadrant, both [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex] are negative.
3. Determine [tex]\(\tan{\theta}\)[/tex]:
[tex]\[
\tan{\theta} = \frac{1}{\cot{\theta}} = \frac{1}{\frac{3}{4}} = \frac{4}{3}
\][/tex]
In the third quadrant, [tex]\(\tan{\theta}\)[/tex] is positive since both the sine and cosine are negative, and their quotient gives a positive value. Therefore:
[tex]\[
\tan{\theta} = \frac{4}{3}
\][/tex]
4. Express [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex] in terms of a common variable:
[tex]\[
\cot{\theta} = \frac{3}{4} = \frac{\cos{\theta}}{\sin{\theta}} \implies \cos{\theta} = 3k \text{ and } \sin{\theta} = 4k
\][/tex]
5. Use the Pythagorean identity:
The Pythagorean identity states:
[tex]\[
\sin^2{\theta} + \cos^2{\theta} = 1
\][/tex]
Substituting [tex]\(\cos{\theta} = 3k\)[/tex] and [tex]\(\sin{\theta} = 4k\)[/tex]:
[tex]\[
(3k)^2 + (4k)^2 = 1
\][/tex]
[tex]\[
9k^2 + 16k^2 = 1
\][/tex]
[tex]\[
25k^2 = 1 \implies k^2 = \frac{1}{25} \implies k = \frac{1}{5}
\][/tex]
6. Find [tex]\(\sin{\theta}\)[/tex] and [tex]\(\cos{\theta}\)[/tex]:
[tex]\[
\sin{\theta} = 4k = 4 \times \frac{1}{5} = \frac{4}{5}
\][/tex]
[tex]\[
\cos{\theta} = 3k = 3 \times \frac{1}{5} = \frac{3}{5}
\][/tex]
Since both sine and cosine are negative in the third quadrant:
[tex]\[
\sin{\theta} = -\frac{4}{5} \quad \text{and} \quad \cos{\theta} = -\frac{3}{5}
\][/tex]
7. Determine [tex]\(\csc{\theta}\)[/tex]:
[tex]\[
\csc{\theta} = \frac{1}{\sin{\theta}} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}
\][/tex]
Now compare these values to the options given:
A. [tex]\(\sin{\theta} = \frac{3}{5}\)[/tex] [tex]\(\rightarrow\)[/tex] Incorrect, it should be [tex]\(-\frac{4}{5}\)[/tex].
B. [tex]\(\csc{\theta} = -\frac{5}{3}\)[/tex] [tex]\(\rightarrow\)[/tex] Incorrect, it should be [tex]\(-\frac{5}{4}\)[/tex].
C. [tex]\(\cos{\theta} = -\frac{3}{5}\)[/tex] [tex]\(\rightarrow\)[/tex] Correct.
D. [tex]\(\tan{\theta} = \frac{4}{3}\)[/tex] [tex]\(\rightarrow\)[/tex] Correct.
Therefore, the correct answers are C and D.
