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Pythagoras Theorem, Trigonometric Ratios, radian and degrees, degrees to radians, Sine rule

rebelmaths

rebel

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i)$\\frac{\\cos(\\var{num1[0]}^{\\circ}) + \\sin(\\var{num1[1]}^{\\circ})}{\\tan(\\var{num1[2]}^{\\circ})} = \\var{ans1}$

ii) $\\frac{\\tan(\\var{num2[0]}^{\\circ}) + \\sin(\\var{num2[1]}^{\\circ})}{\\cos(\\var{num2[2]}^{\\circ}) - \\sin(\\var{num2[3]}^{\\circ})} = \\var{ans2}$

iii) $\\frac{\\cos(\\var{num3[0]}) + \\sin(\\var{num3[1]})}{\\tan(\\var{num3[2]})} = \\var{ans3}$

iv) $\\frac{\\sin(\\var{num4[0]}) + \\cos(\\var{num4[1]})}{\\tan(\\var{num4[2]}) - \\sin(\\var{num4[3]})} = \\var{ans4}$

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$\\frac{\\cos(\\var{num1[0]}^{\\circ}) + \\sin(\\var{num1[1]}^{\\circ})}{\\tan(\\var{num1[2]}^{\\circ})}$

[[0]]

$\\frac{\\tan(\\var{num2[0]}^{\\circ}) + \\sin(\\var{num2[1]}^{\\circ})}{\\cos(\\var{num2[2]}^{\\circ}) - \\sin(\\var{num2[3]}^{\\circ})}$

[[0]]

$\\frac{\\cos(\\var{num3[0]}) + \\sin(\\var{num3[1]})}{\\tan(\\var{num3[2]})}$

[[0]]

$\\frac{\\sin(\\var{num4[0]}) + \\cos(\\var{num4[1]})}{\\tan(\\var{num4[2]}) - \\sin(\\var{num4[3]})}$

[[0]]

Evaluate each of te following, correct to 3 decimal places:

Don't forget the order of operations. BOMDAS!!

Hint: Check mode of calculator, radians mode needed for some and degree mode needed for others!!

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Cos, Sin and Tan

rebelmaths

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$x^2 = \\var{lent1}^2 + \\var{lent2}^2$

$x = \\sqrt(\\var{lent1}^2 + \\var{lent2}^2)$

$x = \\var{ans1}$

$\\var{l21}^2 = x^2 + \\var{l22}^2$

$x^2 = \\var{l21}^2 - \\var{l22}^2$

$x = \\sqrt(\\var{l21}^2 - \\var{l22}^2)$

$x = \\var{ans2}$

$\\var{l32}^2 = x^2 + \\var{l31}^2$

$x^2 = \\var{l32}^2 - \\var{l31}^2$

$x = \\sqrt(\\var{l32}^2 - \\var{l31}^2)$

$x = \\var{ans3}$

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

Write the name of each side in the correct box:

Note: The bottom right angle is a right angle (90$^{\\circ}$).

{tri(30,45,56)}

x = [[0]]

y = [[1]]

z = [[2]]

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The hypotenuse is the side across from the right angle.

The line touching the important angle is called it's adjacent line.

The line opposite the important angle is called it's opposite line.

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Write the name of each side in the correct box:

Note: The bottom left angle is a right angle (90$^{\\circ}$).

{tri2(25,19,16,48)}

x = [[0]]

y = [[1]]

z = [[2]]

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Write the name of each side in the correct box:

Note: The bottom left angle is a right angle (90$^{\\circ}$).

{tri3(10,33,73)}

x = [[0]]

y = [[1]]

z = [[2]]

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Write the name of each side in the correct box:

Note: The top right angle is a right angle (90$^{\\circ}$).

{tri4(60,78,48)}

x = [[0]]

y = [[1]]

z = [[2]]

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Find the value of x:

Note: The bottom right angle is a right angle (90$^{\\circ}$).

{tri5(lent1,lent2,ang1)}

x = [[0]] mm

Find the value of x:

Note: The bottom left angle is a right angle (90$^{\\circ}$).

{tri6(l21,l22,l23,ang2)}

x = [[0]] mm

Find the value of x:

Note: The bottom left angle is a right angle (90$^{\\circ}$).

{tri7(l31,l32,l33)}

x = [[0]] mm

Write Adjacent, Opposite or Hypotenuse in the label answer boxes that correspond to the triangle above them, in parts(a - d).

The angle labelled \"ang\" is the important one in each case.

In parts(e - g), find the length of the unknown side using Pythagorass Theorem.

Solve the following mensuration questions to 2 decimal places:

Note: You may need to scroll down to see the diagrams.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ans1": {"definition": "sqrt(lent1^2+lent2^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "ans2": {"definition": "sqrt(l21^2-l22^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2", "description": ""}, "ans3": {"definition": "sqrt(l32^2-l31^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans3", "description": ""}, "lent1": {"definition": "random(25..35)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent1", "description": ""}, "lent2": {"definition": "random(45..60)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent2", "description": ""}, "l21": {"definition": "random(30..45)", "templateType": "anything", "group": "Ungrouped variables", "name": "l21", "description": ""}, "l23": {"definition": "random(9..15)", "templateType": "anything", "group": "Ungrouped variables", "name": "l23", "description": ""}, "l22": {"definition": "random(12..18)", "templateType": "anything", "group": "Ungrouped variables", "name": "l22", "description": ""}, "l32": {"definition": "random(60..70)", "templateType": "anything", "group": "Ungrouped variables", "name": "l32", "description": ""}, "ang1": {"definition": "random(63..65)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang1", "description": ""}, "ang2": {"definition": "random(48..65)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang2", "description": ""}, "l31": {"definition": "random(30.5..45.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "l31", "description": ""}, "l33": {"definition": "random(40..50)", "templateType": "anything", "group": "Ungrouped variables", "name": "l33", "description": ""}}, "metadata": {"description": "

Pythagoras' Theorem and naming sides of right angled triangle

rebelmaths

i) $\\tan(\\var{ang1}^{\\circ}$) = $\\frac{x}{\\var{lent1}}$

$\\var{lent1} \\times Tan(\\var{ang1}^{\\circ}) = x$

$x = \\var{ans11}mm$

$\\cos(\\var{ang1}$) = $\\frac{\\var{lent1}}{y}$

$y = \\frac{\\var{lent1}}{\\cos(\\var{ang1})}$

$y = \\var{ans12}mm$

ii) Sin($\\var{ang2}^{\\circ}$) = $\\frac{x}{\\var{lent2}}$

$\\var{lent2} \\times \\sin(\\var{ang2}^{\\circ}) = x$

$x = \\var{ans21}mm$

$\\cos(\\var{ang2}^{\\circ}$) = $\\frac{y}{\\var{lent2}}$

$\\var{lent2} \\times \\cos(\\var{ang2}^{\\circ}) = y$

$y = \\var{ans22}mm$

iii) $\\sin(\\var{ang3}^{\\circ}$) = $\\frac{\\var{lent3}}{x}$

$x = \\frac{\\var{lent3}}{\\sin(\\var{ang3}^{\\circ})}$

$x = \\var{ans31}mm$

$\\tan(\\var{ang3}^{\\circ}$) = $\\frac{\\var{lent3}}{y}$

$y = \\frac{\\var{lent3}}{\\tan(\\var{ang3}^{\\circ})}$

$y = \\var{ans32}mm$

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

Find the values of x and y:

Note: The bottom right angle is a right angle (90$^{\\circ}$).

{tri(lent1,ans11,ang1)}

x = [[0]]mm

y = [[1]]mm

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

First name all the sides.

To find x: We have the adjacent and we need the opposite. Which of the trigonometric ratios include adjacent and opposite? Tan.

To find y: We have the adjacent and we need the hypotenuse. Which of the trigonometric ratios include adjacent and hypotenuse?

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "{ans11}", "strictPrecision": false, "minValue": "{ans11}", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "{ans12}", "strictPrecision": false, "minValue": "{ans12}", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

Find the values of x and y:

Note: The bottom left angle is a right angle (90$^{\\circ}$).

{tri2(lent2,ans21,ans22,ang2)}

x = [[0]]mm

y = [[1]]mm

Find the values of x and y:

Note: The bottom left angle is a right angle (90$^{\\circ}$).

{tri3(ans32,lent3,ang3)}

x = [[0]]mm

y = [[1]]mm

Watch the following video for help understanding solving right angle triangle problems.

Video

Solve the following mensuration questions to 2 decimal places:

Note: You may need to scroll down to see the diagrams.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"lent1": {"definition": "random(20..35#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent1", "description": ""}, "lent2": {"definition": "random(20..45#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent2", "description": ""}, "lent3": {"definition": "random(30.5..45.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent3", "description": ""}, "ans12": {"definition": "(lent1/(sin(radians(180-(ang1+90)))))*sin(radians(90))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans12", "description": ""}, "ans11": {"definition": "(lent1/(sin(radians(180-(ang1+90)))))*sin(radians(ang1))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans11", "description": ""}, "ang1": {"definition": "random(55..70)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang1", "description": ""}, "ang2": {"definition": "random(48..65)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang2", "description": ""}, "ang3": {"definition": "random(61..75)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang3", "description": ""}, "ans31": {"definition": "(lent3/(sin(radians(ang3))))*sin(radians(90))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans31", "description": ""}, "ans32": {"definition": "(lent3/(sin(radians(ang3))))*sin(radians(90-ang3))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans32", "description": ""}, "ans22": {"definition": "(lent2/(sin(radians(90))))*sin(radians(180-(ang2+90)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22", "description": ""}, "ans21": {"definition": "(lent2/(sin(radians(90))))*sin(radians(ang2))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21", "description": ""}}, "metadata": {"description": "

Solve for x and y on a given triangle

rebelmaths

i) $\\frac{\\var{lent21}}{\\sin(90^{\\circ})} = \\frac{\\var{lent22}}{\\sin(Y^{\\circ})}$

$\\sin(Y^{\\circ}) = \\var{lent22} \\div \\frac{\\var{lent21}}{\\sin(90^{\\circ})}$

$Y = \\var{ans21}^{\\circ}$

$\\frac{\\var{lent21}}{\\sin(90^{\\circ})} = \\frac{x}{\\sin(90^{\\circ}-\\var{ans21})}$

$x = \\frac{\\var{lent21}}{\\sin(90^{\\circ})} \\times \\sin(90^{\\circ}-\\var{ans21}^{\\circ}) = \\var{ans22}mm$

ii) $y^2 + \\var{lent11}^2 = \\var{lent12}^2 $

$y^2 = \\var{lent12}^2 - \\var{lent11}^2 $

$y = \\sqrt(\\var{lent12}^2 - \\var{lent11}^2) = \\var{ans11}mm$

$\\frac{\\var{lent12}}{\\sin(90^{\\circ})} = \\frac{\\var{ans21}}{\\sin(x^{\\circ})}$

$\\sin(x^{\\circ}) = \\var{ans21} \\div \\frac{\\var{lent12}}{\\sin(90^{\\circ})}$

$x = \\var{ans12}^{\\circ}$

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

Find the values of x and y:

Note:The bottom left angle is a right angle (90$^{\\circ}$).

{tri2(lent21,lent22,ans22)}

x = [[0]]mm

y = [[1]] $^{\\circ}$

To find x: Use Pythagoras Theorem to find the missing side.

To find y: For the most accurate answer use the sides that were given. What trigonometric ratio includes those two sides? Sin. Remember to use inverse-sin to find the angle.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}, {"prompt": "

Find the values of x and y:

Note:The top right angle is a right angle (90$^{\\circ}$).

{tri(lent11,lent12,ans12)}

x = [[0]] $^{\\circ}$

y = [[1]] mm

Solve the following mensuration questions to 2 decimal places:

Note: You may need to scroll down to see the diagrams.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"lent31": {"definition": "random(72..98#2)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent31", "description": ""}, "lent12": {"definition": "random(41..51#2)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent12", "description": ""}, "ang31": {"definition": "random(71..85)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang31", "description": ""}, "ang32": {"definition": "random(47..55)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang32", "description": ""}, "lent11": {"definition": "random(20..28#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent11", "description": ""}, "ans12": {"definition": "90-(degrees(arcsin(lent11/(lent12/(sin(radians(90)))))))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans12", "description": ""}, "ans11": {"definition": "sqrt(lent12^2 - lent11^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans11", "description": ""}, "l2": {"definition": "(ans32/(sin(radians(90))))*sin(radians(180-(ang32+90)))", "templateType": "anything", "group": "Ungrouped variables", "name": "l2", "description": ""}, "lent22": {"definition": "random(35..48)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent22", "description": ""}, "lent21": {"definition": "random(55..75)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent21", "description": ""}, "h3": {"definition": "(ans32/(sin(radians(90))))*sin(radians(ang32))", "templateType": "anything", "group": "Ungrouped variables", "name": "h3", "description": ""}, "ans31": {"definition": "(lent31/(sin(radians(ang31))))*sin(radians(ang32))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans31", "description": ""}, "ans32": {"definition": "(lent31/(sin(radians(ang31))))*sin(radians(180-(ang32+ang31)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans32", "description": ""}, "ans33": {"definition": "0.5*lent31*h3", "templateType": "anything", "group": "Ungrouped variables", "name": "ans33", "description": ""}, "ans22": {"definition": "(lent21/(sin(radians(90))))*sin(radians(90-(ans21)))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22", "description": ""}, "ans21": {"definition": "degrees(arcsin(lent22/(lent21/(sin(radians(90))))))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21", "description": ""}}, "metadata": {"description": "

Find angle and side in a right angled triangle.

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Q4 Solve for x and y on a given triangle and calculate the area", "extensions": [], "custom_part_types": [], "resources": [["question-resources/q4.png", "/srv/numbas/media/question-resources/q4.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}], "functions": {"tri": {"definition": "\nvar c = document.createElement('canvas');\n $(c).attr('width',900).attr('height',900);\n var context = c.getContext('2d');\n\n context.beginPath();\n context.moveTo(300,800);\n context.lineTo((x*8+300),800);\n context.lineTo((w*8+300),(800-(y*8)));\n context.closePath();\n context.stroke();\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = l+'mm';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(300+((8*x)*.85)),790-((y*8)/2));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = 'x';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(280+((8*w)/2)),(800-((y*8)/2)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '12px sans-serif';\n var wstring = a1+'\\xB0';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(270+(8*w)),(855-(y*8)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '12px sans-serif';\n var wstring = a2+'\\xB0';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(230+(8*x)),(790));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = 'y';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(310+((8*x)/2)),820);\n\n \n\n return c;\n ", "type": "html", "parameters": [["x", "number"], ["y", "number"], ["l", "number"], ["a1", "number"], ["a2", "number"], ["w", "number"]], "language": "javascript"}, "tri4": {"definition": "var c = document.createElement('canvas');\n $(c).attr('width',900).attr('height',900);\n var context = c.getContext('2d');\n\n context.beginPath();\n context.moveTo(100,800);\n context.lineTo((l*8+100),800);\n context.lineTo((l2*8+100),(800-(h*8)));\n context.closePath();\n context.stroke();\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = l+'mm';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(((8*l)/2)),(820));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = 'x';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(100+(4*(8*l))/5),(800-((h*8)/2)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = 'y';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(100+((8*l)/5)),(800-((h*8)/2)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '12px sans-serif';\n var wstring = a1 + '\\xB0';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(90+((l2*8))),(845-((h*8))));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '12px sans-serif';\n var wstring = a2 + '\\xB0';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(120),(790));\n\n \n\n return c;\n ", "type": "html", "parameters": [["l", "number"], ["x", "number"], ["y", "number"], ["a1", "number"], ["a2", "number"], ["h", "number"], ["l2", "number"]], "language": "javascript"}, "tri3": {"definition": "var c = document.createElement('canvas');\n $(c).attr('width',900).attr('height',900);\n var context = c.getContext('2d');\n\n context.beginPath();\n context.moveTo(100,800-(y*5));\n context.lineTo((l2*5+100),800);\n context.lineTo((x*5+100),(800-(y*5)));\n context.closePath();\n context.stroke();\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = x+'mm';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(100+((5*x)/2)),(790-(y*5)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = 'x';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(100+((5*l2))/3),(800-((y*5)/2)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = 'y';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,((120+(l2*5))+((5*(x-l2))/2)),(800-((y*5)/2)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '12px sans-serif';\n var wstring = a1 + '\\xB0';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(90+((l2*5))),(770));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '12px sans-serif';\n var wstring = a2 + '\\xB0';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(120),(820-(y*5)));\n\n \n\n return c;\n ", "type": "html", "parameters": [["x", "number"], ["y", "number"], ["a1", "number"], ["a2", "number"], ["l2", "number"]], "language": "javascript"}, "tri2": {"definition": "\nvar c = document.createElement('canvas');\n $(c).attr('width',900).attr('height',900);\n var context = c.getContext('2d');\n\n context.beginPath();\n context.moveTo(200,(800-(y*8)));\n context.lineTo((((x-z)*8)+200),800);\n context.lineTo((200+x*8),(800));\n context.closePath();\n context.stroke();\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '16px sans-serif';\n var wstring = 'x';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(140+((8*x))),(790));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '16px sans-serif';\n var wstring = 'Y';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(240),(840-(y*8)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = l1+'mm';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(200+(8*x)/2),(800-((y*8)/2)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = 'z';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,((200+(x-z)*8)+(z*8)/2),(820));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '20px sans-serif';\n var wstring = l2+'mm';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,210,(800-((y*8)/2)));\n\n //draw labels\n context.fillStyle = '#000';\n context.font = '12px sans-serif';\n var wstring = a+'\\xB0';\n var tw = context.measureText(wstring).width;\n context.fillText(wstring,(210+(x-z)*8),(790));\n\n \n return c;\n ", "type": "html", "parameters": [["x", "number"], ["y", "number"], ["z", "number"], ["l1", "number"], ["l2", "number"], ["a", "number"]], "language": "javascript"}}, "ungrouped_variables": ["lent11", "ans11", "ans12", "lent22", "lent21", "ans21", "ang31", "lent31", "ans31", "ang32", "ans32", "h3", "ans33", "l2", "ang21", "ang11", "ang12", "h1", "w1", "ans13", "ans22", "ans23", "h2", "w2", "ans24a", "ans24"], "tags": ["rebelmaths"], "advice": "

i) $\\frac{\\var{lent11}}{\\sin(180^{\\circ}-(\\var{ang11}^{\\circ}+\\var{ang12}^{\\circ}))} = \\frac{x}{\\sin(\\var{ang12}^{\\circ})}$

$x = \\frac{\\var{lent11}}{\\sin(180^{\\circ}-(\\var{ang11}^{\\circ}+\\var{ang12}^{\\circ}))} \\times \\sin(\\var{ang12}^{\\circ}) = \\var{ans11}mm$

$\\frac{\\var{lent11}}{\\sin(180^{\\circ}-(\\var{ang11}^{\\circ}+\\var{ang12}^{\\circ}))} = \\frac{y}{Sin(\\var{ang11}^{\\circ})}$

$y = \\frac{\\var{lent11}}{Sin(180^{\\circ}-(\\var{ang11}^{\\circ}+\\var{ang12}^{\\circ}))} \\times \\sin(\\var{ang11}^{\\circ}) = \\var{ans12}mm$

$\\frac{1}{2} \\times \\var{ans11} \\times \\var{lent11} \\times \\sin(\\var{ang11}^{\\circ}) = \\var{ans13}mm^2$

ii) $\\frac{\\var{lent21}}{\\sin(\\var{ang21}^{\\circ})} = \\frac{\\var{lent22}}{\\sin(x^{\\circ})}$

$\\sin(x^{\\circ}) = \\var{lent22} \\div \\frac{\\var{lent21}}{\\sin(\\var{ang21}^{\\circ})}$

$x = \\var{ans21}^{\\circ}$

$y = 180^{\\circ} - (\\var{ans21}^{\\circ}+\\var{ang21}^{\\circ}) = \\var{ans22}^{\\circ}$

$\\frac{\\var{lent21}}{\\sin(\\var{ang21}^{\\circ})} = \\frac{z}{\\sin(\\var{ans22}^{\\circ})}$

$z = \\frac{\\var{lent21}}{\\sin(\\var{ang21})} \\times \\sin(\\var{ans22}) = \\var{ans23}mm$

$\\frac{1}{2} \\times \\var{ans23} \\times \\var{lent22} \\times \\sin(\\var{ang21}^{\\circ}) = \\var{ans24}mm^2$

iii) $\\frac{\\var{lent31}}{\\sin(\\var{ang31}^{\\circ})} = \\frac{x}{\\sin(180^{\\circ}-(\\var{ang31}^{\\circ}+\\var{ang32}^{\\circ}))}$

$x = \\frac{\\var{lent31}}{\\sin(\\var{ang31}^{\\circ})} \\times \\sin(180^{\\circ}-(\\var{ang31}^{\\circ}+\\var{ang32}^{\\circ})) = \\var{ans31}mm$

$\\frac{\\var{lent31}}{\\sin(\\var{ang31}^{\\circ})} = \\frac{y}{\\sin(\\var{ang32}^{\\circ})}$

$y = \\frac{\\var{lent31}}{\\sin(\\var{ang31}^{\\circ})} \\times \\sin(\\var{ang32}^{\\circ}) = \\var{ans32}mm$

$\\frac{1}{2} \\times \\var{ans31} \\times \\var{ans32} \\times \\sin(\\var{ang31}^{\\circ}) = \\var{ans33}mm^2$

iv) $\\frac{\\var{lent31a}}{\\sin(\\var{ang31a}^{\\circ})} = \\frac{x}{\\sin(\\var{ang32a}^{\\circ})}$

$x = \\frac{\\var{lent31a}}{\\sin(\\var{ang31a}^{\\circ})} \\times \\sin(\\var{ang32a}^{\\circ}) = \\var{ans31a}mm$

$\\frac{\\var{lent31a}}{\\sin(\\var{ang31a}^{\\circ})} = \\frac{y}{\\sin(180^{\\circ}-(\\var{ang32a}^{\\circ}+\\var{ang31a}^{\\circ}))}$

$y = \\frac{\\var{lent31a}}{\\sin(\\var{ang31a})} \\times \\sin(180-(\\var{ang32a}+\\var{ang31a})) = \\var{ans32a}mm$

$\\frac{1}{2} \\times \\var{ans31a} \\times \\var{ans32a} \\times \\sin(\\var{ang31a}^{\\circ}) = \\var{ans33a}mm^2$

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

Find the values of x and y:

Note: Keep answers to 3 decimal places for accuracy.

{tri(ans12,h1,lent11,ang11,ang12,w1)}

x = [[0]]mm

y = [[1]]mm

Area = [[2]]$mm^2$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans11}+0.5", "minValue": "{ans11}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans12}+0.5", "minValue": "{ans12}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans13}+1", "minValue": "{ans13}-1", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "

First you need to find the third angle. Remember that the three angles of any triangle add up to 180 degrees. Next, pair up opposite sides and angles to use the sine rule.

Find the values of x and y:

Note: Keep answers to 3 decimal places for accuracy.

{tri2(w2,h2,ans23,lent21,lent22,ang21)}

x = [[0]] $^{\\circ}$

y = [[1]] $^{\\circ}$

z = [[2]]mm

Area = [[3]]$mm^2$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans21}+0.5", "minValue": "{ans21}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans22}+0.5", "minValue": "{ans22}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans23}+0.5", "minValue": "{ans23}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans24}+1", "minValue": "{ans24}-1", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

Find the values of x and y:

Note: Keep answers to 3 decimal places for accuracy.

{tri3(lent31,h3,ang31,ang32,l2)}

x = [[0]]mm

y = [[1]]mm

Area = [[2]]$mm^2$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans31}+0.5", "minValue": "{ans31}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans32}+0.5", "minValue": "{ans32}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans33}+1", "minValue": "{ans33}-1", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "

Find the values of x and y:

Note: Keep answers to 3 decimal places for accuracy.

{tri4(lent31a,ans31a,ans32a,ang31a,ang32a,h3a,l2a)}

x = [[0]]mm

y = [[1]]mm

Area = [[2]]$mm^2$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans31a}+0.5", "minValue": "{ans31a}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans32a}+0.5", "minValue": "{ans32a}-0.5", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "2", "maxValue": "{ans33a}+1", "minValue": "{ans33a}-1", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

Watch the video below to better understand the sine rule formula

Video

Solve the following mensuration questions to 2 decimal places:

Note: You may need to scroll down to see the diagrams.

", "variable_groups": [{"variables": ["ang31a", "lent31a", "ans31a", "ang32a", "ans32a", "h3a", "ans33a", "l2a"], "name": "Q4"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"l2a": {"definition": "precround((ans32a/(sin(radians(90))))*sin(radians(180-(ang32a+90))),2)", "templateType": "anything", "group": "Q4", "name": "l2a", "description": ""}, "h3a": {"definition": "precround((ans32a/(sin(radians(90))))*sin(radians(ang32a)),2)", "templateType": "anything", "group": "Q4", "name": "h3a", "description": ""}, "h2": {"definition": "precround((lent21/(sin(radians(90))))*sin(radians(ans21)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "h2", "description": ""}, "h3": {"definition": "precround((ans32/(sin(radians(90))))*sin(radians(180-(ang31+ang32))),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "h3", "description": ""}, "h1": {"definition": "precround((lent11/(sin(radians(90))))*sin(radians(ang12)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "h1", "description": ""}, "w2": {"definition": "precround((lent21/(sin(radians(90))))*sin(radians(90-ans21)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "w2", "description": ""}, "w1": {"definition": "precround((ans12-(lent11/(sin(radians(90))))*sin(radians(90-ang12))),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "w1", "description": ""}, "ans24": {"definition": "precround((0.5*lent22*ans23*sin(radians(ang21))),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans24", "description": ""}, "ans23": {"definition": "precround((lent21/(sin(radians(ang21))))*sin(radians(ans22)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans23", "description": ""}, "ans22": {"definition": "180-(ans21+ang21)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22", "description": ""}, "ans21": {"definition": "precround(degrees(arcsin(lent22*sin(radians(ang21))/lent21)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21", "description": ""}, "ang21": {"definition": "random(120..140)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang21", "description": ""}, "lent31": {"definition": "random(101..120)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent31", "description": ""}, "lent31a": {"definition": "random(72..98#2)", "templateType": "anything", "group": "Q4", "name": "lent31a", "description": ""}, "lent11": {"definition": "random(35..45)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent11", "description": ""}, "ans32a": {"definition": "precround((lent31a/(sin(radians(ang31a))))*sin(radians(180-(ang32a+ang31a))),2)", "templateType": "anything", "group": "Q4", "name": "ans32a", "description": ""}, "ans24a": {"definition": "precround((0.5*(w2)*h2),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans24a", "description": ""}, "ans12": {"definition": "precround((lent11/(sin(radians(180-(ang11+ang12)))))*sin(radians(ang11)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans12", "description": ""}, "ans13": {"definition": "precround((0.5*ans12*h1),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans13", "description": ""}, "ans11": {"definition": "precround((lent11/(sin(radians(180-(ang11+ang12)))))*sin(radians(ang12)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans11", "description": ""}, "lent22": {"definition": "random(35..50)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent22", "description": ""}, "lent21": {"definition": "random(75..85)", "templateType": "anything", "group": "Ungrouped variables", "name": "lent21", "description": ""}, "ans31": {"definition": "precround((lent31/(sin(radians(ang31))))*sin(radians(180-(ang31+ang32))),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans31", "description": ""}, "ans32": {"definition": "precround((lent31/(sin(radians(ang31))))*sin(radians(ang32)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans32", "description": ""}, "ans33": {"definition": "precround((0.5*lent31*h3),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans33", "description": ""}, "ang31a": {"definition": "random(71..85)", "templateType": "anything", "group": "Q4", "name": "ang31a", "description": ""}, "ang12": {"definition": "random(55..(ang11-8))", "templateType": "anything", "group": "Ungrouped variables", "name": "ang12", "description": ""}, "ang32a": {"definition": "random(47..55)", "templateType": "anything", "group": "Q4", "name": "ang32a", "description": ""}, "ang11": {"definition": "random(63..76)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang11", "description": ""}, "ang31": {"definition": "random(75..85)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang31", "description": ""}, "ang32": {"definition": "random(47..57)", "templateType": "anything", "group": "Ungrouped variables", "name": "ang32", "description": ""}, "ans31a": {"definition": "precround((lent31a/(sin(radians(ang31a))))*sin(radians(ang32a)),2)", "templateType": "anything", "group": "Q4", "name": "ans31a", "description": ""}, "l2": {"definition": "precround((ans31/(sin(radians(90))))*sin(radians(90-ang32)),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "l2", "description": ""}, "ans33a": {"definition": "precround((0.5*lent31a*h3a),2)", "templateType": "anything", "group": "Q4", "name": "ans33a", "description": ""}}, "metadata": {"description": "

Solve for x and y on a given triangle and calculate the area

rebelmaths

To convert degrees to radians: multiply by $\\pi$ and divide by 180.

$\\frac{\\var{d1} \\times \\pi}{180} = \\var{a1}$

$\\frac{\\var{d2} \\times \\pi}{180} = \\var{a2}$

$\\frac{\\var{d3} \\times \\pi}{180} = \\var{a3}$

$\\frac{(\\var{d41}^{\\circ}\\var{d42}^{\\prime}) \\times \\pi}{180} = \\var{a4}$

$\\frac{(\\var{d51}^{\\circ}\\var{d52}^{\\prime}) \\times \\pi}{180} = \\var{a5}$

$\\frac{(\\var{d61}^{\\circ}\\var{d62}^{\\prime}) \\times \\pi}{180} = \\var{a6}$

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

$\\var{d1}^{\\circ}$ = [[0]] radians

To convert to radians divide by 180 and multiply by $\\pi$

$\\var{d2}^{\\circ}$ = [[0]] radians

$\\var{d3}^{\\circ}$ = [[0]] radians

$\\var{d41}^{\\circ}$ $\\var{d42}^{\\prime}$ = [[0]] radians

$\\var{d51}^{\\circ}$ $\\var{d52}^{\\prime}$ = [[0]] radians

$\\var{d61}^{\\circ}$ $\\var{d62}^{\\prime}$ = [[0]] radians

Convert the following to radians:

Round your answer to 2 decimal places.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"d2": {"definition": "random(65..87#0.01)", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "d3": {"definition": "random(515..599)", "templateType": "anything", "group": "Ungrouped variables", "name": "d3", "description": ""}, "d51": {"definition": "random(62..75)", "templateType": "anything", "group": "Ungrouped variables", "name": "d51", "description": ""}, "a1": {"definition": "radians(d1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a3": {"definition": "radians(d3)", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "a2": {"definition": "radians(d2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "a5": {"definition": "radians(d51+(d52/60))", "templateType": "anything", "group": "Ungrouped variables", "name": "a5", "description": ""}, "a4": {"definition": "radians(d41+(d42/60))", "templateType": "anything", "group": "Ungrouped variables", "name": "a4", "description": ""}, "a6": {"definition": "radians(d61+(d62/60))", "templateType": "anything", "group": "Ungrouped variables", "name": "a6", "description": ""}, "d42": {"definition": "random(40..55)", "templateType": "anything", "group": "Ungrouped variables", "name": "d42", "description": ""}, "d41": {"definition": "random(15..37)", "templateType": "anything", "group": "Ungrouped variables", "name": "d41", "description": ""}, "d1": {"definition": "random(15..37)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "d61": {"definition": "random(101..114)", "templateType": "anything", "group": "Ungrouped variables", "name": "d61", "description": ""}, "d52": {"definition": "random(15..27)", "templateType": "anything", "group": "Ungrouped variables", "name": "d52", "description": ""}, "d62": {"definition": "random(32..47)", "templateType": "anything", "group": "Ungrouped variables", "name": "d62", "description": ""}}, "metadata": {"description": "

Converting Degrees to Radians

rebelmaths

To convert degrees to radians: multiply by 180 and divide by $\\pi$.

$\\frac{\\var{d1} \\times 180}{\\pi} = \\var{a1}$

$\\frac{\\var{d2} \\times 180}{\\pi} = \\var{a2}$

$\\frac{\\var{d3} \\times 180}{\\pi} = \\var{a3}$

$\\frac{\\var{d4}\\pi}{5} \\times \\frac{180}{\\pi} = \\var{a4}$

$\\frac{\\var{d5}\\pi}{5} \\times \\frac{180}{\\pi} = \\var{a5}$

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

$\\var{d1}$ radians= [[0]] $^{\\circ}$

To convert from radians to degrees, multiply by 180 and divide by $\\pi$

$\\var{d2}$ radians= [[0]] $^{\\circ}$

$\\var{d3}$ radians= [[0]] $^{\\circ}$

$\\frac{\\var{d4}\\pi}{5}$ radians= [[0]] $^{\\circ}$

$\\frac{\\var{d5}\\pi}{5}$ radians= [[0]] $^{\\circ}$

Convert the following to degrees:

Round your answer to 2 decimal places.

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Converting Radians to Degrees

rebelmaths

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