Using the given information to complete the equation $y= A \\cos{ \\left( \\frac{2 \\pi}{P} x \\right) }+V $ that describes an electromagnetic wave and calculating the smallest angle, $x$, for which $y=y_0$.
", "licence": "None specified"}, "statement": "The shape of an electromagnetic wave is modelled by the equation:
$y=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)}+\\var{v}$
where $x$ is measured in radians.
", "advice": "The equation
\n\\[ \\begin{split} y=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)}+\\var{v} \\end{split} \\]
\nis a wave function.
\nThe general form of a wave function can be written as:
\n\\[ \\begin{split} y=A\\cos{\\left(\\frac{2 \\pi}{P}\\left(x-H\\right)\\right)}+V \\end{split} \\]
\nWhere $V$ is the average value, $A$ is the amplitute, $H$ the phase and $P$ the wavelenght (also called period).
\nBy comparying the equation with the general form we can notice that:
\na) The amplitude of the wave is $A=\\var{a}$
\nb)The period can be found by solving the equation
\n\\[ \\begin{split} \\frac{\\pi}{\\simplify{{p}/2}} &= \\frac{2 \\pi}{P} \\\\ \\pi P&= \\simplify{{p}/2}\\times 2\\pi \\\\ \\pi P&= {p}\\pi \\\\ P&=\\frac{{p} \\pi}{\\pi} \\\\ P &=\\var{p} \\end{split} \\]
\nSo, the wavelenght is $P=\\var{p}$.
\nTo find the smallest value of $x$ for which $y=\\var{y1}$. We need to substitute $y=\\var{y1}$ in the equation and solve for $x$.
\n\\[ \\begin{split} \\var{y1}&=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)}+\\var{v} \\\\ \\var{y1}-\\var{v} &=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)} \\\\ \\var{y1-v} &=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)} \\\\ \\frac{\\var{y1-v}}{\\var{a}} &=\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)}\\end{split} \\]
\nWe take $\\cos^{-1}$ from both sides:
\n\\[ \\begin{split} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)&=\\frac{\\pi x}{\\simplify{{p}/2}} \\\\ \\simplify{{p}/2} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)&=\\pi x \\\\ \\frac{\\simplify{{p}/2} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)}{\\pi}&=x \\end{split} \\]
\nWe can use the calculator to find that $x= \\frac{\\simplify{{p}/2} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)}{\\pi}=\\var{ansC}$ radians (rounded to 2 decimal places).
\nWe can use the calculator to find that $x= \\frac{\\simplify{{p}/2} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)}{\\pi}=\\var{ansC}$ radians.
\nWrite down the amplitude of this wave.
\n$A=$[[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "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": "Calculate the wavelength of this wave.
\nThe wavelength is [[0]].
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{p}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "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": "Find the smallest value of $x$ for which $y=\\var{y1}$.
\n$x=$[[0]] radians.
\nGive your answer rounded to 2 decimal places, if needed.
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