// Numbas version: exam_results_page_options
{"name": "Exact values for csc, sec, cot (acute, degrees)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/exact_values.svg", "/srv/numbas/media/question-resources/exact_values.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Exact values for csc, sec, cot (acute, degrees)", "tags": [], "metadata": {"description": "<p>multiple choice testing csc, sec, cot&nbsp;of random(30, 45, 60) degrees</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Often we prefer to work with&nbsp;exact values rather than approximations from a calculator.</p>", "advice": "<p>Recall that $\\csc\\theta=\\dfrac{1}{\\sin\\theta}$, $\\sec\\theta=\\dfrac{1}{\\cos\\theta}$, and&nbsp;$\\cot\\theta=\\dfrac{1}{\\tan\\theta}$.</p>\n<p></p>\n<p>By drawing the following triangles we can determine the exact values of $\\sin$, $\\cos$ and $\\tan$ (and their reciprocals $\\csc$, $\\sec$, $\\cot$) for the angles $30^\\circ$, $45^\\circ$ and $60^\\circ$.</p>\n<p><br/><img src=\"resources/question-resources/exact_values.svg\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Alternatively, one can memorise the following table:&nbsp;</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">$30^\\circ$</td>\n<td style=\"text-align: center;\">$45^\\circ$</td>\n<td style=\"text-align: center;\">$60^\\circ$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\sin$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\cos$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\tan$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{3}}$</td>\n<td style=\"text-align: center;\">$1$</td>\n<td style=\"text-align: center;\">$\\sqrt{3}$</td>\n</tr>\n</tbody>\n</table>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"theta": {"name": "theta", "group": "Ungrouped variables", "definition": "random(30,45,60)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["theta"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The exact value of $\\csc(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$2$</p>", "<p>$\\sqrt{2}$</p>", "<p>$\\dfrac{2}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$1$</p>"], "matrix": ["if(theta=30,1,0)", "if(theta=45,1,0)", "if(theta=60,1,0)", 0, 0, 0], "distractors": ["", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The exact value of $\\sec(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$2$</p>", "<p>$\\sqrt{2}$</p>", "<p>$\\dfrac{2}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$1$</p>"], "matrix": ["if(theta=60,1,0)", "if(theta=45,1,0)", "if(theta=30,1,0)", 0, 0, 0], "distractors": ["", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The exact value of $\\cot(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$2$</p>", "<p>$\\sqrt{2}$</p>", "<p>$\\dfrac{2}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$1$</p>"], "matrix": ["0", "0", "0", "if(theta=30,1,0)", "if(theta=60,1,0)", "if(theta=45,1,0)"], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}