«cos. «a b. cos. cos a + cos b = 2cos 2. cos. cos a cos b = 2sin 2. EXAMPLE 1 (by working on the right hand side) Prove the following identity

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1 Sec. 09 notes Famous IDs: Sum & Products Identities Main Idea We continue to expand the list of very famous trigonometric identities, and to practice our proving skills. Virtually all identities presented in the last section, this section and in the next few sections, ultimately, come from from the mother of them all. Product-to-Sums Identities sin a sin b = [(a b (a + b] a b = [(a b + (a + b] sin a b = [sin(a + b + sin(a b] Sums-to-Products Identities a + b sin a + sin b = sin a b sin a sin b = sin a + b a + b = a + b a b = sin a b a + b a b a b sin EXAMPLE (by working on the right hand side Prove the following identity sin a sinb = [(a b (a + b] We will work on the right hand side using the identities we have already proven for (a b and (a+b sin a sinb? = [(a b (a + b] want to know [ab + sin a sinb (ab sinasin b] proven identities [ab + sin a sinb ab + sin a sin b] distribute the negative [ sin a sinb] clean up = sin a sin b kachin! kachin! c hands pg.

2 Sec. 09 notes EXAMPLE (by tweaking a known identity, the identity from example one. Prove the following identity x + y x y x y = sin sin We begin with a know identity identity, the one proven above. We then rearrange the equation to match the idenity we are trying to prove. The essential step is the substitution, the real tweak, substituting a = ( x+y and b = ( x y. [( x + y + sinasin b = [(a b (a + b] (known, proven above sinasin b = (a b (a + b (mult by, a little tweak (a + b (a b = sinasinb (move sides, another little tweak ( ] [] x y x + y x y x + y x y = sin sin (the real tweak, sub. x + y x y x y = sin sin (clean up & done!! c hands pg.

3 Famous IDs: Sum & Products Identities. Prove and OWN everyone of these famous identities. sin a sinb = [(a b (a + b] We will work on the right hand side using the identities we have already proven for (a b and (a + b sin a sinb? = [(a b (a + b] want to know [ab + sin a sinb (ab sinasin b] proven identities [ab + sin a sinb ab + sin a sin b] distribute the negative [ sin a sinb] clean up = sin a sin b kachin! kachin!. Prove and OWN everyone of these famous identities. a + b a b a b = sin sin We begin with a know identity identity, the one proven above. We then rearrange the equation to match the idenity we are trying to prove. The essential step is the substitution, the real tweak, substituting a = ( x+y and b = ( x y. [( x + y + (a + b (a b = sinasinb ( ] [] ( x y x + y x y x + y = sin sinasin b = [(a b (a + b] (known, proven above sinasin b = (a b (a + b (mult by, a little tweak ( x + y x y = sin (move sides, another little tweak ( x y (the real tweak, sub. ( x y (clean up & done!! sin sin 3. Prove and OWN everyone of these famous identities. ab = [(a b + (a + b] c hands pg. 3

4 We will work on the right hand side using the identities we have already proven for (a b and (a + b ab? = [(a b + (a + b] want to know [ab + sinasin b + ab sin a sinb] proven identities [ab + sinasin b ab sin a sinb] distribute the negative [ a b] clean up = a b kachin! kachin! 4. Prove and OWN everyone of these famous identities. a + b a b a + b = We begin with a know identity identity, the one proven above. We then rearrange the equation to match the idenity we are trying to prove. The essential step is the substitution, the real tweak, substituting a = ( ( x+y and b = x y. [( x + y + [(a b + (a + b] = ab (known, proven above (a b + (a + b = ab (mult by, a little tweak (a + b + (a b = ab (prepare for substitution.. ( ] [] x y x + y x y x + y x y + = (the real tweak, sub. x + y x y x + y = (clean up & done!! 5. Prove and OWN everyone of these famous identities. sin a b = [sin(a + b + sin(a b] see solutions to ex. #,, 3, &4 6. Prove and OWN everyone of these famous identities. a + b a b sin a + sin b = sin see solutions to ex. #,, 3, &4 c hands pg. 4

5 7. Prove and OWN everyone of these famous identities. a b a + b sin a sin b = sin see solutions to ex. #,, 3, &4 8. Prove the following non-famous identity. (4x(3x = x + (7x ab = [(a b + (a + b] (already proven the tweak, sub a = 4x and b = 3x 4x3x = [(4x 3x + (4x + 3x] (sub 4x3x = (x + (7x (algebra 9. Prove the following non-famous identity. (4x + (3x = (3.5x (.5x ab = [(a b + (a + b] (already proven the tweak, sub a = 3.5x and b =.5x 3.5x.5x = [(3.5x.5x + (3.5x +.5x] (sub 3.5x.5x = (3x + (4x (algebra 0. Prove the following non-famous identity. sin(4x + sin(6x = sin(5x(x sin a b = [sin(a + b + sin(a b] (already proven the tweak, sub a = 5x and b = x sin 5xx = [sin(5x + x + sin(5x x] (sub sin5xx = sin(6x + sin(4x (algebra c hands pg. 5

6 . Prove the following non-famous identity. ( (x (3x (7x = x First note (a = a is a proven and famous identity, sub a = x to recognize the variation, (x = (4x... then.. work on left side... ( (x (3x (7x = (4x(3x (7x now from ex. #8 we see that (4x(3x = x + (7x therefore... ( (x (3x (7x = (4x(3x (7x (con t.. = x + (7x (7x (sub = x. Without calculators determine if the following is true, then explain = (37.5 (7.5 ab = [(a b + (a + b] (already proven the tweak, sub a = 37.5 and b = = [( ( ] (sub = (30 + (45 (algebra = + (algebra 3. (**Prove the following non-famous identity. 4 (x(3x 3x (7x = x work on left side... notice... then continue as in ex. # 4 (x(3x 3x (7x = [ (x ] (3x (7x c hands pg. 6