Question

\tan(a) = -\frac{15}{8} \text{ and } \cos(a) < 0. \text{ find } \sin(a) \text{ and } \cos(a).

\bigcirc \sin(a) = \frac{15}{17}, \cos(a) = -\frac{8}{17}
\bigcirc \sin(a) = \frac{15}{17}, \cos(a) = \frac{8}{17}
\bigcirc \sin(a) = \frac{8}{17}, \cos(a) = -\frac{15}{17}
\bigcirc \sin(a) = -\frac{15}{17}, \cos(a) = -\frac{8}{17}

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Pythagorean Theorem",
"Right Triangle Trigonometry"
],
"new_concepts": [
"Trigonometric Signs in Quadrants"
],
"current_concepts": [
"Pythagorean Theorem",
"Right Triangle Trigonometry",
"Trigonometric Signs in Quadrants"
]
}
</pre_analysis>

<reasoning>

Determine the quadrant of angle A

Given:
\[
\tan(A) = -\frac{15}{8} < 0 \quad \text{and} \quad \cos(A) < 0
\]
Since \(\tan(A) < 0\) and \(\cos(A) < 0\), the angle \(A\) must lie in Quadrant II.
In Quadrant II, \(\sin(A) > 0\).

Calculate the hypotenuse and trigonometric ratios

Let the reference right triangle have opposite side \(y = 15\) and adjacent side \(x = -8\).
Using the Pythagorean theorem:
\[
r = \sqrt{x^2 + y^2} = \sqrt{(-8)^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17
\]
Thus:
\[
\sin(A) = \frac{y}{r} = \frac{15}{17}
\]
\[
\cos(A) = \frac{x}{r} = -\frac{8}{17}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(\sin(A) = \frac{15}{17}, \cos(A) = -\frac{8}{17}\)</mcq-correct>
<mcq-option>(B) \(\sin(A) = \frac{15}{17}, \cos(A) = \frac{8}{17}\)</mcq-option>
<mcq-option>(C) \(\sin(A) = \frac{8}{17}, \cos(A) = -\frac{15}{17}\)</mcq-option>
<mcq-option>(D) \(\sin(A) = -\frac{15}{17}, \cos(A) = -\frac{8}{17}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Signs in Quadrants"
]
}
</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": [
"Pythagorean Theorem",
"Right Triangle Trigonometry"
],
"new_concepts": [
"Trigonometric Signs in Quadrants"
],
"current_concepts": [
"Pythagorean Theorem",
"Right Triangle Trigonometry",
"Trigonometric Signs in Quadrants"
]
}
</pre_analysis>

<reasoning>

Determine the quadrant of angle A

Given:
\[
\tan(A) = -\frac{15}{8} < 0 \quad \text{and} \quad \cos(A) < 0
\]
Since \(\tan(A) < 0\) and \(\cos(A) < 0\), the angle \(A\) must lie in Quadrant II.
In Quadrant II, \(\sin(A) > 0\).

Calculate the hypotenuse and trigonometric ratios

Let the reference right triangle have opposite side \(y = 15\) and adjacent side \(x = -8\).
Using the Pythagorean theorem:
\[
r = \sqrt{x^2 + y^2} = \sqrt{(-8)^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17
\]
Thus:
\[
\sin(A) = \frac{y}{r} = \frac{15}{17}
\]
\[
\cos(A) = \frac{x}{r} = -\frac{8}{17}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(\sin(A) = \frac{15}{17}, \cos(A) = -\frac{8}{17}\)</mcq-correct>
<mcq-option>(B) \(\sin(A) = \frac{15}{17}, \cos(A) = \frac{8}{17}\)</mcq-option>
<mcq-option>(C) \(\sin(A) = \frac{8}{17}, \cos(A) = -\frac{15}{17}\)</mcq-option>
<mcq-option>(D) \(\sin(A) = -\frac{15}{17}, \cos(A) = -\frac{8}{17}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Signs in Quadrants"
]
}
</post_analysis>