To solve the given problem, we need to find the appropriate trigonometric values based on the information provided:
- [tex]\(\cot \theta = \frac{3}{4}\)[/tex]
- Angle [tex]\(\theta\)[/tex] is in the third quadrant
Let's break it down step-by-step:
1. Identify the cotangent value and related trigonometric values:
Since [tex]\(\cot \theta = \frac{3}{4}\)[/tex], we know that [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex].
Thus, [tex]\(\cos \theta = 3k\)[/tex] and [tex]\(\sin \theta = 4k\)[/tex] for some proportional constant [tex]\(k\)[/tex].
2. Determine the signs in the third quadrant:
In the third quadrant, both sine and cosine values are negative. Therefore, [tex]\(\cos \theta\)[/tex] and [tex]\(\sin \theta\)[/tex] must be negative.
3. Find the hypotenuse using the Pythagorean Theorem:
[tex]\[
\cos^2 \theta + \sin^2 \theta = 1
\][/tex]
Substitute the expressions [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]\[
25k^2 = 1 \implies k^2 = \frac{1}{25} \implies k = \pm \frac{1}{5}
\][/tex]
Since we are in the third quadrant and both trigonometric functions are negative:
[tex]\[
k = -\frac{1}{5}
\][/tex]
4. Calculate [tex]\(\cos \theta\)[/tex] and [tex]\(\sin \theta\)[/tex]:
[tex]\[
\cos \theta = 3k = 3 \left(-\frac{1}{5}\right) = -\frac{3}{5}
\][/tex]
[tex]\[
\sin \theta = 4k = 4 \left(-\frac{1}{5}\right) = -\frac{4}{5}
\][/tex]
5. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3}
\][/tex]
Therefore,
[tex]\[
\tan \theta = \frac{4}{3}
\][/tex]
which corresponds to option A.
6. Verify [tex]\(\csc \theta\)[/tex] and [tex]\(\sin \theta\)[/tex]:
[tex]\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}
\][/tex]
- Correct choice does not correspond to C.
And the value of [tex]\(\sin \theta\)[/tex] already calculated as [tex]\(-\frac{4}{5}\)[/tex], which does not match with D.
So, summarizing the choices:
- A. [tex]\(\tan \theta = \frac{4}{3}\)[/tex] is correct.
- B. [tex]\(\cos \theta = -\frac{3}{5}\)[/tex] is correct.
- C. [tex]\(\csc \theta = -\frac{5}{4}\)[/tex] is correct.
- D. [tex]\(\sin \theta = \frac{3}{5}\)[/tex] is incorrect as it should be [tex]\(\sin \theta = -\frac{4}{5}\)[/tex].
Thus, the correct answers should be:
A. [tex]\(\tan \theta = \frac{4}{3}\)[/tex]
B. [tex]\(\cos \theta = -\frac{3}{5}\)[/tex]
C. [tex]\(\csc \theta = -\frac{5}{4}\)[/tex]
