5.1 Angles in Standard Position

5.1 Standard Angles

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Photo by Hindrik S.

http://www.flickr.com/photos/63991153@N00/6516894031

Unit5.

Trigonometry

5.1 Standard Angles

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NOT PART OF THIS COURSE

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5.1 & 5.2 ­ Definitions of Angles (Intro to Grade 12 Trig)

5.3 & 5.4 ­ Sine & Cosine Laws

5.1 Standard Angles

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Unit5.TrigonometryMentalMath

Setup thethreetrigonometricratiosfor θ.

θ

3

4

5

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5.1AnglesinStandardPosition

Anangleisformedbytworayswithacommonendpoint.

Anglesareoftenrepresentedas rotationsofaray.

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AnglesinStandardPosition

Anangleissaidtobeanangleinstandardpositionifitsvertexisattheoriginofacoordinategridanditsinitialarmcoincideswiththepositivex‐axis.

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5.1 Standard Angles

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E.g.Sketcheachangleinstandardposition.Statethequadrantinwhichtheterminalarmlies:

a.)36° b.)210° c.)315°

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ReferenceAngles

Foreachangleinstandardposition,thereisacorrespondingacuteanglecalledthereferenceangle.

Thereferenceangleistheacuteangleformedbetweentheterminalarmandthex‐axis.

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EgDeterminethereferenceangle θR foreachangle. Sketchinstandardpositionandlabelthereferenceangle.

a.)130° a.)300°

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Whataretheanglesinstandardpositionwithareferenceangleof20°?

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E.g.Determinetheangleinstandardpositionwhenanangleof40°isrelectedinthe:a.)inthey‐axisb.)inthex‐axisc.)inthey‐axisandtheninthex‐axis

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E.g.Determinetheangleinstandardpositionwhenanangleof40°isrelectedinthe:a.)inthey‐axisb.)inthex‐axisc.)inthey‐axisandtheninthex‐axis

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E.g.Determinetheangleinstandardpositionwhenanangleof40°isrelectedinthe:a.)inthey‐axisb.)inthex‐axisc.)inthey‐axisandtheninthex‐axis

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SpecialRightTriangles

Thereare3specialanglesinmathematics‐30°,45°,60°.

Theirtrigonometricratioshave exactvalues.E.g.Let'sindtheexactvalueofa45°‐45°‐90°trianglewithasidelengthof1unit.

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E.g.Findthetrigonometricratiosforthefollowingtriangle

1

2√3

30°

60°

sin60°=

cos60°=

tan60°=

sin30°=

cos30°=

tan30°=

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E.g.Allieislearningtoplaythepiano.Herteacherusesametronometohelpherkeeptime.Thependulumarmofthemetronomeis10cmlong.Foroneparticulartempo,thesettingresultsinthearmmovingbackandforthfromastartpositionof60°to120°.Whathorizontaldistancedoesthetipofthearmmoveinonebeat?Giveanexactanswer.

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Homeworkp83­84#1­6`