// Numbas version: finer_feedback_settings
{"name": "Trigonometry: Alternative Trigonometric Forms 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trigonometry: Alternative Trigonometric Forms 3", "tags": [], "metadata": {"description": "<p>Rewriting a trigonometric expression of the form $A\\cos(\\theta)-B\\sin(\\theta)$ to $R\\cos(\\theta+\\alpha)$ by calculating $R$ and $\\alpha$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If \\[\\simplify[unitFactor]{{A}cos(theta)}-\\simplify[unitFactor]{{B}sin(theta)} = R \\cos (\\theta + \\alpha),\\]</p>\n<p>find the values for $R$ and $\\alpha$, given $R&gt;0$ and $0&lt;\\alpha&lt;\\frac{\\pi}{2}$.</p>", "advice": "<p>To find $R$ and $\\alpha$ we want to first rewrite our equation using the double-angle formula, $\\cos(a+b)=\\cos(a)\\cos(b)-\\sin(a)\\sin(b)$:</p>\n<p>\\[ \\begin{split}\\simplify[unitFactor]{{A}cos(theta)}-\\simplify[unitFactor]{{B}sin(theta)} &amp;\\,= R \\cos (\\theta + \\alpha) \\\\ &amp;\\,= R(\\cos(\\theta)\\cos(\\alpha) - \\sin(\\theta)\\sin(\\alpha))&nbsp;\\\\ &amp;\\,= R\\cos(\\theta)\\cos(\\alpha) - R\\sin(\\theta)\\sin(\\alpha). \\end{split} \\]</p>\n<p>By comparing the coefficients of $\\cos(\\theta)$ and $\\sin(\\theta)$, we find that</p>\n<p>\\[ R\\cos(\\alpha) = \\var{A},\\quad \\text{and} \\quad R\\sin(\\alpha) = \\var{B}. \\]</p>\n<p>To calculate $R$, we want to square these results and add them together, allowing us to make use of $\\sin^2(\\alpha)+\\cos^2(\\alpha) = 1$:</p>\n<p>{Rsol}</p>\n<p>Similarly, to find $\\alpha$ we can divide $R\\sin(\\alpha) = \\var{B}$ by $R\\cos(\\alpha) = \\var{A}$, and use the identity $\\tan(\\alpha) = \\frac{\\sin(\\alpha)}{\\cos(\\alpha)}$:</p>\n<p>\\[ \\frac{R\\sin(\\alpha)}{R\\cos(\\alpha)} = \\frac{\\var{B}}{\\var{A}} \\implies \\tan(\\alpha) = \\simplify[fractionNumbers]{{B/A}}.\\]</p>\n<p>Therefore, \\[ \\begin{split} \\alpha &amp;\\,= \\tan^{-1}\\left(\\simplify[fractionNumbers]{{B/A}}\\right) \\\\ &amp;\\,= \\var{alpharound} \\text{ (2 d.p.)}. \\end{split} \\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(1..5 except A)", "description": "", "templateType": "anything", "can_override": false}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "sqrt(A^2+B^2)", "description": "", "templateType": "anything", "can_override": false}, "Rround": {"name": "Rround", "group": "Ungrouped variables", "definition": "precround(R,2)", "description": "", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Ungrouped variables", "definition": "arctan(B/A)", "description": "", "templateType": "anything", "can_override": false}, "Rsol": {"name": "Rsol", "group": "Ungrouped variables", "definition": "if(R=round(R),'{Rsol1}','{Rsol2}')", "description": "", "templateType": "anything", "can_override": false}, "Rsol1": {"name": "Rsol1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split} R^2\\\\cos^2(\\\\alpha) + R^2 \\\\sin^2(\\\\alpha) &amp;\\\\,= \\\\var{A}^2+\\\\var{B}^2 \\\\\\\\ R^2 (\\\\cos^2(\\\\alpha) +\\\\sin^2(\\\\alpha)) &amp;\\\\,= \\\\var{A^2+B^2} \\\\\\\\ R &amp;\\\\,= \\\\var{R}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "Rsol2": {"name": "Rsol2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split} R^2\\\\cos^2(\\\\alpha) + R^2 \\\\sin^2(\\\\alpha) &amp;\\\\,= \\\\var{A}^2+\\\\var{B}^2 \\\\\\\\ R^2 (\\\\cos^2(\\\\alpha) +\\\\sin^2(\\\\alpha)) &amp;\\\\,= \\\\var{A^2+B^2} \\\\\\\\ R &amp;\\\\,=&nbsp;\\\\sqrt{\\\\var{A^2+B^2}}\\\\\\\\ &amp;\\\\,=\\\\var{Rround} \\\\text{ (2 d.p.)}.&nbsp;\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "alpharound": {"name": "alpharound", "group": "Ungrouped variables", "definition": "precround(alpha,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "R", "Rround", "alpha", "alpharound", "Rsol", "Rsol1", "Rsol2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": 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