#4. if $\sin \theta = \frac{5}{13}$ and $\theta$ is in q2, find $\cos \theta$ and $\tan \theta$.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Trigonometric Quadrants", "Trigonometric Ratios of General Angles" ], "new_concepts": [], "current_concepts": [ "Trigonometric Quadrants", "Trigonometric Ratios of General Angles", "Reference Angles" ] } ## Identify coordinates in Quadrant II $$ \begin{aligned} &y = 5,\quad r = 13\ &x^2 + y^2 = r^2 \implies x^2 + 5^2 = 13^2 \implies x^2 = 144\ &\theta \in \text{Q2} \implies x < 0 \implies x = -12 \end{aligned} $$ ## Calculate trigonometric ratios $$ \begin{aligned} &\cos\theta = \frac{x}{r} = -\frac{12}{13}\ &\tan\theta = \frac{y}{x} = -\frac{5}{12} \end{aligned} $$ $\cos\theta = -\frac{12}{13}$ and $\tan\theta = -\frac{5}{12}$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Ratios of General Angles" ] }

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Identify coordinates in Quadrant II

Calculate trigonometric ratios

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