![](\"resources/question-resources/exact_values_radians_0wJX122.svg\"/)
testing sin, cos, tan of random(0,90,120,135,150,180,210,225,240,270,300,315,330) degrees but in radians
"}, "advice": "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 $\\dfrac{\\pi}{6}$, $\\dfrac{\\pi}{4}$ and $\\dfrac{\\pi}{3}$.
\n
Alternatively, one can memorise the following table:
| \n | $\\dfrac{\\pi}{6}$ | \n$\\dfrac{\\pi}{4}$ | \n$\\dfrac{\\pi}{3}$ | \n
| \n | \n | \n | \n |
| $\\sin$ | \n$\\dfrac{1}{2}$ | \n$\\dfrac{1}{\\sqrt{2}}$ | \n$\\dfrac{\\sqrt{3}}{2}$ | \n
| \n | \n | \n | \n |
| $\\cos$ | \n$\\dfrac{\\sqrt{3}}{2}$ | \n$\\dfrac{1}{\\sqrt{2}}$ | \n$\\dfrac{1}{2}$ | \n
| \n | \n | \n | \n |
| $\\tan$ | \n$\\dfrac{1}{\\sqrt{3}}$ | \n$1$ | \n$\\sqrt{3}$ | \n
That combined with the unit circle definitions:
\nand some understanding of congruent triangles:
\n\n\nallows us to work out $\\sin$, $\\cos$ and $\\tan$ for certain angles regardless of what quadrant the point is in. Because whatever angle we are asked about, we can always use the triangle in the first quadrant to determine the side lengths and then consider the signs of the coordinates separately.
\n\nFor example, to determine $\\sin\\left(\\frac{7\\pi}{6}\\right)$, $\\cos\\left(\\frac{7\\pi}{6}\\right)$ and $\\tan\\left(\\frac{7\\pi}{6}\\right)$ we first draw the following:
\n![](\"resources/question-resources/unit_circle_working_radians.svg\")
From this diagram, we can see that $\\cos\\left(\\frac{7\\pi}{6}\\right)=-\\cos\\left(\\frac{\\pi}{6}\\right)$, and $\\sin\\left(\\frac{7\\pi}{6}\\right)=-\\sin\\left(\\frac{\\pi}{6}\\right)$ since the triangles are congruent and we are in the 3rd quadrant where both the $x$ and $y$ values (and hence the $\\cos$ and $\\sin$ values) are negative.
\nBut given we know these exact values, we can conclude \\[\\cos\\left(\\frac{7\\pi}{6}\\right)=-\\cos\\left(\\frac{\\pi}{6}\\right)=-\\dfrac{\\sqrt{3}}{2},\\] \\[\\sin\\left(\\frac{7\\pi}{6}\\right)=-\\sin\\left(\\frac{\\pi}{6}\\right)=-\\dfrac{1}{2},\\] and finally \\[\\tan\\left(\\frac{7\\pi}{6}\\right)=\\dfrac{\\sin\\left(\\frac{7\\pi}{6}\\right)}{\\cos\\left(\\frac{7\\pi}{6}\\right)}=\\dfrac{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}=\\dfrac{1}{\\sqrt{3}}.\\]
", "variables": {"theta": {"name": "theta", "templateType": "anything", "definition": "random([0,0],[pi/2,'\\$\\\\frac{\\\\pi}{2}\\$'],[2pi/3,'\\$\\\\frac{2\\\\pi}{3}\\$'],[3pi/4,'\\$\\\\frac{3\\\\pi}{4}\\$'],[5pi/6,'\\$\\\\frac{5\\\\pi}{6}\\$'],[pi,'\\$\\\\pi\\$'],[7pi/6,'\\$\\\\frac{7\\\\pi}{6}\\$'],[5pi/4,'\\$\\\\frac{5\\\\pi}{4}\\$'],[4pi/3,'\\$\\\\frac{4\\\\pi}{3}\\$'],[3pi/2,'\\$\\\\frac{3\\\\pi}{2}\\$'],[5pi/3,'\\$\\\\frac{5\\\\pi}{3}\\$'],[7pi/4,'\\$\\\\frac{7\\\\pi}{4}\\$'],[11pi/6,'\\$\\\\frac{11\\\\pi}{6}\\$'])", "description": "", "group": "Ungrouped variables"}}, "name": "Maria's copy of Exact values for sin, cos, tan (0 to 2pi, radians)", "rulesets": {}, "functions": {}, "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "ungrouped_variables": ["theta"], "statement": "Often we prefer to work with exact values rather than approximations from a calculator. In this question we require you input your answer without decimals and without entering the words sin, cos or tan. For example, to input the exact value of $\\sin\\left(\\frac{\\pi}{3}\\right)$, which is $\\dfrac{\\sqrt{3}}{2}$, you would input sqrt(3)/2
", "parts": [{"marks": 0, "prompt": "The exact value of $\\sin\\left(\\var{theta[1]}\\right)$ is [[0]].
\nThe exact value of $\\cos\\left(\\var{theta[1]}\\right)$ is [[1]].
\nIs $\\tan\\left(\\var{theta[1]}\\right)$ defined? [[2]]
\nThe exact value of $\\tan\\left(\\var{theta[1]}\\right)$ is [[3]].
\nYes
", "No
"]}, {"marks": 1, "expectedvariablenames": [], "checkingtype": "absdiff", "type": "jme", "showpreview": true, "checkvariablenames": false, "showFeedbackIcon": true, "answer": "{tan({theta[0]})}", "variableReplacements": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "notallowed": {"message": "", "strings": [".", "sin", "cos", "tan", "cosec", "sec", "cot"], "showStrings": true, "partialCredit": 0}, "scripts": {}, "showCorrectAnswer": false, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "scripts": {"constructor": {"order": "after", "script": "// override the default 'submit' method for the part\n// we only want to submit gap 3 if the answer to gap 2 is \"yes\"\nthis.submit = Numbas.util.extend(function() {\n this.gaps[0].submit();\n this.gaps[1].submit();\n this.gaps[2].submit();\n if(this.gaps[2].yes) {\n this.gaps[3].submit();\n } else {\n this.gaps[3].answered = true;\n }\n},Part.prototype.submit);"}, "mark": {"order": "instead", "script": "var gaps = this.gaps;\n// if the student said \"yes\" to gap 2, then mark their factorisation\nif(gaps[2].yes) {\n this.__proto__.mark.apply(this);\n// otherwise, work out the credit based on the amounts awarded for the first three gaps\n} else {\n this.setCredit((gaps[0].credit*gaps[0].marks+gaps[1].credit*gaps[1].marks+gaps[2].credit*gaps[2].marks)/(gaps[0].marks+gaps[1].marks+gaps[2].marks));\n}"}}, "showFeedbackIcon": true, "showCorrectAnswer": false, "type": "gapfill"}], "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}