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Sum/Difference Angle Identities It will be helpful to have a filled in Unit Circle to refer to as you work through this lesson! To use these identities, we need to think of two angles on the Unit Circle that we could add or subtract in order to obtain 15 o . One possibility is 45 o minus 30 o . You also could have used 60 o minus 45 o or any combination of Unit Circle angles that would yield 15 o . This work just illustrates one possibility. The answer would be the same regardless. Since we are subtracting, we use the Difference Identity for cosine. We use the identity, plug in the Unit Circle values, and then simplify. Notice how the signs between the terms differ depending on if you are using the sum or difference version of the identity. Once we obtain radian values that sum to !!" !# , we apply the Sum Identity for tangent, plug in the Unit Circle values, and then simplify.

Sum/Difference Angle Identities It will be helpful to have a …...Sum/Difference Angle Identities It will be helpful to have a filled in Unit Circle to refer to as you work through

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Page 1: Sum/Difference Angle Identities It will be helpful to have a …...Sum/Difference Angle Identities It will be helpful to have a filled in Unit Circle to refer to as you work through

Sum/DifferenceAngleIdentities

ItwillbehelpfultohaveafilledinUnitCircletorefertoasyouworkthroughthislesson!

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

Noticehowthesignsbetweenthetermsdifferdependingonifyouareusingthesumordifferenceversionoftheidentity.

Onceweobtainradianvaluesthatsumto!!"

!#,we

applytheSumIdentityfortangent,plugintheUnitCirclevalues,andthensimplify.

Page 2: Sum/Difference Angle Identities It will be helpful to have a …...Sum/Difference Angle Identities It will be helpful to have a filled in Unit Circle to refer to as you work through

SampleAnswer:

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

ßoptionofanglestoaddtogether ßAnswer.

Inthesetwo“proofs”,westartbyexpandingthesumordifferenceusingtheappropriateidentity.ThenwesubstituteUnitCircleValuesfortheradians.Thesimplifiedexpressionshouldmatchthedesiredresult.

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