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Finn funksjoner som er periodiske

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Notasjon: For å skrive inn rottegn, feks roten av 5, skriv sqrt(5)

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I denne oppgaven skal du tegne opp enhetssirkelen, repetere definisjonen på sinus og cosinus og regne ut verdiene for de oppgitte vinklene for hånd. Da trenger du litt kunnskap fra ungdomsskolen: pythagoras setning, egenskaper ved trekanter med vinklene 30, 60 og 90 og ved likebeinte trekanter.

\n

Fyll in tabellen med sinus- og cosinus verdiene til de oppgitte vinklene: 

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$v$$0^{\\circ}$$30^{\\circ}$$45^{\\circ}$$60^{\\circ}$$90^{\\circ}$$180^{\\circ}$$270^{\\circ}$$360^{\\circ}$
$\\sin(v)$[[0]][[1]][[2]][[3]][[4]][[5]][[6]][[7]]
$\\cos(v)$[[8]][[9]][[10]][[11]][[12]][[13]][[14]][[15]]
\n

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"implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"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": "sqrt(3)/2", "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": []}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], 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Bruk enhetssirkelen og skriv sinus og cosinus verdiene av disse vinklene i form av spisse vinkler

\n

{geogebra_applet('https://www.geogebra.org/m/gqba3v8x',defs2)}

\n

$\\sin(\\var{b}^{\\circ})=\\sin($[[0]]$^{\\circ})$

\n

$\\sin(\\var{b}^{\\circ})=\\cos($[[1]]$^{\\circ})$

\n

$\\cos(\\var{b}^{\\circ})=-\\cos($[[2]]$^{\\circ})$

\n

$\\cos(\\var{b}^{\\circ})=-\\sin($[[3]]$^{\\circ})$

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a", "maxValue": "a", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{90-a}", "maxValue": "{90-a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a", "maxValue": "a", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{90-a}", "maxValue": "{90-a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "

Bruk enhetssirkelen og skriv sinus og cosinus verdiene av disse vinklene i form av spisse vinkler

\n

{geogebra_applet('https://www.geogebra.org/m/gqba3v8x',defs3)}

\n

$\\sin(\\var{c}^{\\circ})=-\\sin($[[0]]$^{\\circ})$

\n

$\\sin(\\var{c}^{\\circ})=-\\cos($[[1]]$^{\\circ})$

\n

$\\cos(\\var{c}^{\\circ})=-\\cos($[[2]]$^{\\circ})$

\n

$\\cos(\\var{c}^{\\circ})=-\\sin($[[3]]$^{\\circ})$

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a", "maxValue": "a", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{90-a}", "maxValue": "{90-a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a", "maxValue": "a", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{90-a}", "maxValue": "{90-a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "

Bruk enhetssirkelen og skriv sinus og cosinus verdiene av disse vinklene i form av spisse vinkler

\n

{geogebra_applet('https://www.geogebra.org/m/gqba3v8x',defs4)}

\n

$\\sin(\\var{d}^{\\circ})=-\\sin($[[0]]$^{\\circ})$

\n

$\\sin(\\var{d}^{\\circ})=-\\cos($[[1]]$^{\\circ})$

\n

$\\cos(\\var{d}^{\\circ})=\\cos($[[2]]$^{\\circ})$

\n

$\\cos(\\var{d}^{\\circ})=\\sin($[[3]]$^{\\circ})$

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Bruk summeformlene for å finne nøyaktige sinus- og cosinusverdier 

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$\\sin(105^{\\circ})$

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$\\cos(15^{\\circ})$

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$\\cos(75^{\\circ})$

", "answer": "(sqrt(6)-sqrt(2))/4", "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": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Rettvinklet trekant, sin, cos, tan", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

Det er gitt en rettvinklet trekant

\n

{geogebra_applet('https://www.geogebra.org/m/zuu62zns',defs)}

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Finn nøyaktig verdi av hypotenusen $BC$

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Finn $\\sin\\angle ABC$

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Finn $\\cos\\angle ABC$

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Finn $\\tan\\angle ABC$

", "answer": "{(c-2)/(b-1)}", "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": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Sin, cos, arealsetning", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

Gitt en trekant $ABC$ med sidene $AC=\\var{AC}$, $AB=\\var{AB}$ og vinkel $\\angle BAC=\\var{angleA}^{\\circ}$

\n

{geogebra_applet('https://www.geogebra.org/m/sxxas4f3',defs)}

\n

(Trykk på bildet for å flytte på trekanten.)

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Bruk cosinussetningen og finn siden $BC$

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Bruk sinussetningen for å finne vinkelen $\\angle ABC$ i grader

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Bruk arealsetningen for å finne arealet til trekanten $ABC$ 

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Gjør følgende vinkler om til grader.

\n

(Ser du ikke hele tabellen, trykk Ctrl og - (eller +) for å zoome inn og ut.)

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Finn uten bruk av kalkulator. (Hint: For å skrive inn $\\pi$ bruk pi.)

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$\\arctan 1$

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$\\arctan 0$

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$\\arctan\\sqrt{3}$

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$\\arcsin(-\\frac{1}{2})$

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$\\arcsin 1$

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$\\arccos\\frac{\\sqrt{3}}{2}$

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$\\arccos(-\\frac{\\sqrt{2}}{2})$

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$\\arccos(-1)$

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