To solve this question, let's start with the given value of [tex]\(\tan \theta = \frac{3}{4}\)[/tex] and the information that the terminal point determined by [tex]\(\theta\)[/tex] is in quadrant 3.
1. Understanding [tex]\(\tan \theta = \frac{3}{4}\)[/tex]:
- Tangent of an angle is the ratio of the opposite side to the adjacent side in a right triangle.
- Thus, we have:
[tex]\[ \text{opposite} = 3 \][/tex]
[tex]\[ \text{adjacent} = 4 \][/tex]
2. Finding the hypotenuse:
- Using the Pythagorean theorem, we calculate the hypotenuse of the right triangle:
[tex]\[ \text{hypotenuse} = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} \][/tex]
Substituting the values:
[tex]\[ \text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]
3. Finding [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] in quadrant 3:
- In quadrant 3, both sine and cosine functions are negative.
- Sine of an angle is the ratio of the opposite side to the hypotenuse:
[tex]\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \][/tex]
Since we are in quadrant 3:
[tex]\[ \sin \theta = -\frac{3}{5} \][/tex]
- Cosine of an angle is the ratio of the adjacent side to the hypotenuse:
[tex]\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \][/tex]
Since we are in quadrant 3:
[tex]\[ \cos \theta = -\frac{4}{5} \][/tex]
4. Finding [tex]\(\cot \theta\)[/tex]:
- Cotangent is the reciprocal of tangent:
[tex]\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \][/tex]
5. Finding [tex]\(\csc \theta\)[/tex]:
- Cosecant is the reciprocal of sine:
[tex]\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} \][/tex]
Now, let's match these results with the given options:
- Option A: [tex]\(\sin \theta = \frac{3}{5}\)[/tex] is incorrect because [tex]\(\sin \theta = -\frac{3}{5}\)[/tex] in quadrant 3.
- Option B: [tex]\(\cot \theta = \frac{4}{3}\)[/tex] is correct.
- Option C: [tex]\(\cos \theta = -\frac{4}{5}\)[/tex] is correct.
- Option D: [tex]\(\csc \theta = -\frac{5}{3}\)[/tex] is correct.
Therefore, the correct options are:
B. [tex]\(\cot \theta = \frac{4}{3}\)[/tex]
C. [tex]\(\cos \theta = -\frac{4}{5}\)[/tex]
D. [tex]\(\csc \theta = -\frac{5}{3}\)[/tex]
