Pre-Calc Exam Notes 82

Pre-Calc Exam Notes 82 - with the sum-to-product formulas...

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82 Chapter 3 Identities §3.4 3.4 Other Identities Though the identities in this section fall under the category of “other”, they are perhaps (along with cos 2 θ + sin 2 θ = 1) the most widely used identities in practice. It is very common to encounter terms such as sin A + sin B or sin A cos B in calculations, so we will now derive identities for those expressions. First, we have what are often called the product-to- sum formulas : sin A cos B = 1 2 (sin ( A + B ) + sin ( A B )) (3.37) cos A sin B = 1 2 (sin ( A + B ) sin ( A B )) (3.38) cos A cos B = 1 2 (cos ( A + B ) + cos ( A B )) (3.39) sin A sin B = − 1 2 (cos ( A + B ) cos ( A B )) (3.40) We will prove the ±rst formula; the proofs of the others are similar (see Exercises 1-3). We see that sin ( A + B ) + sin ( A B ) = (sin A cos B + cos A sin B ) + (sin A cos B cos A sin B ) = 2 sin A cos B , so formula (3.37) follows upon dividing both sides by 2. Notice how in each of the above identities a product (e.g. sin A cos B ) of trigonometric functions is shown to be equivalent to a sum (e.g. 1 2 (sin ( A + B ) + sin ( A B ))) of such functions. We can go in the opposite direction,
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Unformatted text preview: with the sum-to-product formulas : sin A + sin B = 2 sin 1 2 ( A + B ) cos 1 2 ( A − B ) (3.41) sin A − sin B = 2 cos 1 2 ( A + B ) sin 1 2 ( A − B ) (3.42) cos A + cos B = 2 cos 1 2 ( A + B ) cos 1 2 ( A − B ) (3.43) cos A − cos B = − 2 sin 1 2 ( A + B ) sin 1 2 ( A − B ) (3.44) These formulas are just the product-to-sum formulas rewritten by using some clever sub-stitutions: let x = 1 2 ( A + B ) and y = 1 2 ( A − B ). Then x + y = A and x − y = B . For example, to derive formula (3.43), make the above substitutions in formula (3.39) to get cos A + cos B = cos ( x + y ) + cos ( x − y ) = 2 · 1 2 (cos ( x + y ) + cos ( x − y )) = 2 cos x cos y (by formula (3.39)) = 2 cos 1 2 ( A + B ) cos 1 2 ( A − B ) . The proofs of the other sum-to-product formulas are similar (see Exercises 4-6)....
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