Sum/Difference Angle Identities It will be helpful to have a …...Sum/Difference Angle Identities...

Sum/DifferenceAngleIdentities

ItwillbehelpfultohaveafilledinUnitCircletorefertoasyouworkthroughthislesson!

Tousetheseidentities,weneedtothinkoftwoanglesontheUnitCirclethatwecouldaddorsubtractinordertoobtain15o.Onepossibilityis45ominus30o.Youalsocouldhaveused60ominus45ooranycombinationofUnitCircleanglesthatwouldyield15o.Thisworkjustillustratesonepossibility.Theanswerwouldbethesameregardless.Sincewearesubtracting,weusetheDifferenceIdentityforcosine.Weusetheidentity,plugintheUnitCirclevalues,andthensimplify.

Noticehowthesignsbetweenthetermsdifferdependingonifyouareusingthesumordifferenceversionoftheidentity.

Onceweobtainradianvaluesthatsumto!!"

!#,we

applytheSumIdentityfortangent,plugintheUnitCirclevalues,andthensimplify.

SampleAnswer:

cos(𝑥 + 𝜋) = 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝜋 − 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝜋= 𝑐𝑜𝑠𝑥 ∙ (−1) − 𝑠𝑖𝑛𝑥 ∙ (0)= −𝑐𝑜𝑠𝑥

ßoptionofanglestoaddtogether ßAnswer.

Inthesetwo“proofs”,westartbyexpandingthesumordifferenceusingtheappropriateidentity.ThenwesubstituteUnitCircleValuesfortheradians.Thesimplifiedexpressionshouldmatchthedesiredresult.