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Unit 4 Section 7

# Unit 4 Section 7 - 4.7 SUM AND DIFFERENCE IDENTITIES Sum...

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4.7 SUM AND DIFFERENCE IDENTITIES Sum and Difference Identities Let and be real numbers. Then 1) 2) 3) Proof of 1): Let us consider two real numbers and . Using the wrapping function , these two real numbers determine two terminal points on the unit circle, say and . In particular, without loss of generality, let and . The arc length is . See figure below. Using the distance formula, we can write an expression for the distance : . This can be simplified as follows: Now let us rotate the arc whose length is in such a way that it will be in standard position. See figure below:

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Although the coordinates of point are now , the distance from to has not changed. Again, by the distance formula, . This can be simplified as follows: . Equating the two equations for , we obtain Solving for , we have . But , so we have the equation . Proof of : The cosine sum formula follows easily from the one we have just derived. Let us consider , this is equal to and by the identity we have proven, we have .
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Unit 4 Section 7 - 4.7 SUM AND DIFFERENCE IDENTITIES Sum...

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