#3. a point $p(4,-3)$ lies on the terminal arm of an angle $\theta$ in standard position. determine the exact trigonometric ratios for the three primary trigonometry ratios.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Trigonometric Quadrants" ], "new_concepts": [ "Trigonometric Ratios of General Angles", "Terminal Arm" ], "current_concepts": [ "Trigonometric Quadrants", "Trigonometric Ratios of General Angles", "Terminal Arm" ] } ## Find the hypotenuse r $$ \begin{aligned} &x = 4,\quad y = -3\ &r = \sqrt{x^2 + y^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5 \end{aligned} $$ ## Calculate the three primary trigonometric ratios $$ \begin{aligned} &\sin\theta = \frac{y}{r} = -\frac{3}{5}\ &\cos\theta = \frac{x}{r} = \frac{4}{5}\ &\tan\theta = \frac{y}{x} = -\frac{3}{4} \end{aligned} $$ $$ \begin{aligned} \sin\theta &= -\frac{3}{5} \ \cos\theta &= \frac{4}{5} \ \tan\theta &= -\frac{3}{4} \end{aligned} $$ { "elements": [ { "type": "line", "params": [[-6, 0], [6, 0]], "properties": { "strokeColor": "#cccccc", "strokeWidth": 1, "fixed": true } }, { "type": "line", "params": [[0, -6], [0, 6]], "properties": { "strokeColor": "#cccccc", "strokeWidth": 1, "fixed": true } }, { "type": "point", "params": [[0, 0]], "properties": { "name": "O", "color": "#583C87", "size": 3, "fixed": true } }, { "type": "point", "params": [[4, -3]], "properties": { "name": "P(4, -3)", "color": "#8C55F2", "size": 4, "fixed": true } }, { "type": "segment", "params": [[0, 0], [4, -3]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "r = 5", "withLabel": true, "labelPosition": [2, -1.2] } }, { "type": "segment", "params": [[4, 0], [4, -3]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2 } }, { "type": "segment", "params": [[0, 0], [4, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "dash": 2 } }, { "type": "text", "params": [2, 0.3, "x = 4"] }, { "type": "text", "params": [4.2, -1.5, "y = -3"] } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Ratios of General Angles" ] }

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Find the hypotenuse r

Calculate the three primary trigonometric ratios

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