Question

the terminal side of an angle in standard position passes through \\(p(-3, -4)\\). what is the value of \\(\tan \theta\\)?

\\(\tan \theta = -\frac{4}{3}\\)

\\(\tan \theta = -\frac{3}{4}\\)

\\(\tan \theta = \frac{3}{4}\\)

\\(\tan \theta = \frac{4}{3}\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Trigonometric Ratios on Unit Circle"
],
"new_concepts": [],
"current_concepts": [
"Trigonometric Ratios on Unit Circle",
"Trigonometric Signs"
]
}
</pre_analysis>

<reasoning>

Identify coordinates of the point on the terminal side

\[
x = -3, \quad y = -4
\]

Apply the trigonometric ratio formula for tangent

\[
\tan\theta = \frac{y}{x}
\]

Substitute the coordinates and simplify

\[
\tan\theta = \frac{-4}{-3} = \frac{4}{3}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\tan\theta = -\frac{4}{3}\)</mcq-option>
<mcq-option>(B) \(\tan\theta = -\frac{3}{4}\)</mcq-option>
<mcq-option>(C) \(\tan\theta = \frac{3}{4}\)</mcq-option>
<mcq-correct>(D) \(\tan\theta = \frac{4}{3}\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Ratios on Unit Circle"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Trigonometric Ratios on Unit Circle"
],
"new_concepts": [],
"current_concepts": [
"Trigonometric Ratios on Unit Circle",
"Trigonometric Signs"
]
}
</pre_analysis>

<reasoning>

Identify coordinates of the point on the terminal side

\[
x = -3, \quad y = -4
\]

Apply the trigonometric ratio formula for tangent

\[
\tan\theta = \frac{y}{x}
\]

Substitute the coordinates and simplify

\[
\tan\theta = \frac{-4}{-3} = \frac{4}{3}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\tan\theta = -\frac{4}{3}\)</mcq-option>
<mcq-option>(B) \(\tan\theta = -\frac{3}{4}\)</mcq-option>
<mcq-option>(C) \(\tan\theta = \frac{3}{4}\)</mcq-option>
<mcq-correct>(D) \(\tan\theta = \frac{4}{3}\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Ratios on Unit Circle"
]
}
</post_analysis>