Famous IDs: Sum-Angle Identities

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1 07 notes Famous IDs: Sum-Angle Identities Main Idea We continue to expand the list of very famous trigonometric identities, and to practice our proving skills. We now prove the second most famous/most important trigonometric identity. By tweaking this identity, we will derive many, many more famous and important trigonometric identities. In some sense, this identity, cos(a b) cosacosb+sinasinb, is the Mother Of Them All. We are now ready to learn it as well as all the famous variations listed below. Sum-Angle Identities the M.O.T.A. cos(a+b) cosacosb sinasinb sin(a+b) sinacosb+cosasinb sin(a b) sinacosb cosasinb tan(a+b) tana+tanb tan(a b) tana tanb 1+tanatanb EXAMPLE 1 Mother Of Them All (by starting with something amazingly creative) Prove the following identity Solution: In order to prove this one, we need several key ingredients. First, we need to be well versed in finding distances between two points. That is, we need to know that the distance squared, d, between two points, (A,B) and (C,K) is given by d (A C) + (B K). Next, we need to be well versed in finding the x and the y coordinates for a reference triangle where the hypothenuse is r and the angle is θ. Finally, we need an amazingly creative idea, apparently unrelated at first, but obviously divinely inspired once it delivers the goods. preliminaries: review distance review coordinates distance by pyth. -6 (4,6) 6 d 8 4 (8,) d (8 4) +(6 ) 6 (?,?) 4 r (rcosθ,rsinθ) θ 4 x y cosθ x r sinθ y r, thus -6 x rcosθ and y rsinθ pg. 1

2 07 notes the brilliant idea: hold this triangle up, then walk away... the brilliant idea: hold this triangle up, then walk away... Now, the idea is to calculate the distance, d, in both cases, before the triangle falls and after it has fallen. (cosa,sina) a 1 b 1 d (cosb,sinb) a b (cos(a b), sin(a b)) d (1,0) d (cosa cosb) +(sina sinb) (from dist. formula) d cos a cosacosb+cos b+sin a sinasinb+sin b (FOIL, alg) d cos a+sin a+cos b+sin b cosacosb sinasinb (to use Pythagoras ID, alg) d 1+1 cosacosb sinasinb (use Pythagoras ID, alg) d cosacosb sinasinb (alg) d [cos(a b) 1] +[sin(a b) 0] (from dist. formula) d cos (a b) cos(a b)+1+sin (a b) (FOIL, alg) d cos (a b)+sin (a b) cos(a b)+1 (to use Pythagoras ID, alg) d 1 cos(a b)+1 d cos(a b) (use Pythagoras ID, alg) (alg) Now, we conclude the square of the distance, d is the same before and after the triangle has fallen. Thus,... d d cosacosb sinasinb cos(b a) cosacosb sinasinb cos(b a) cosacosb+sinasinb cos(b a) (from above) (from work above) (algebra) (algebra) yip-kaei-yeah!! pg.

3 07 notes EXAMPLE Famous Sum-Angle Id. (by tweaking a known Id.) Prove the following identity cos(x+y) cosxcosy sinxsiny Solution: We begin with the MOTA Identity, which we have just proven. Since it is true for all angles a and b, we can apply it for a x and b y. After these values are substituted, we clean it up and voila! cos[x ( y)] cosxcos( y)+sinxsin( y) cos(x+y) cosxcosy +sinxsin( y) cos(x+y) cosxcosy +(sinx)( siny) cos(x+y) cosxcosy sinxsiny Known Id, MOTA the Tweak, substitute a x and b y clean up, cosine is an even function, see Famous Ids clean up, sine is odd function, see Famous Ids clean up, QED EXAMPLE 3 Sum-Angle for Sine (by tweaking a known Id.) Prove the following identity sin(a b) sinacosb cosasinb Solution: we will tweak a known identity. Recall also the Co-Function identities. The main difference between a sine and a cosine is the co for co-mplimentary angles, thus replacing any angle θ with 90 θ results in the co-function ratio. With this in mind we will substitute x (90 a) and y b cos(x+y) cosxcosy sinxsiny cos[(90 a)+b] cos(90 a)cosb sin(90 a)sinb cos[90 (a b)] cos(90 a)cosb sin(90 a)sinb sin[(a b)] sinacosb cosasinb sin(a b) sinacosb cosasinb known ID the tweak, sub algebra co-function Ids goods delivered! famous Id! pg. 3

4 Famous IDs: Sum-Angle Identities 1. Prove and OWN everyone of these famous identities. Sum-Angle Identities the M.O.T.A. cos(a+b) cosacosb sinasinb sin(a+b) sinacosb+cosasinb sin(a b) sinacosb cosasinb tan(a+b) tana+tanb tan(a b) tana tanb 1+tanatanb Solution: 1. To prove, drop the triangle... see video lectures.. done on video lectures.. also see above.. done above.. note once proven, this can be used subsequently on the rest of the identities... To prove cos(x+y) cosxcosy sinxsiny Start with MOTA the tweak it.. sub a x and b y 3. To prove sin(x+y) sinxcosy+cosxsiny Start with MOTA the tweak it.. sub a x and b y+ π... you will need to use some of the previous identities.. such as the co-identities.. and you will need to NOT be afraid.. 4. To prove sin(x y) sinxcosy cosxsiny Start with MOTA the tweak it.. sub a x and b y + π... you will need to use some of the previous identities.. such as the co-identities.. and you will need to NOT be afraid.. OR... use the previous identity.. with x a and y b 5. To prove tan(a+b) tana+tanb, work on the left side.. tan(a+b)? sin(a+b) cos(a+b) sinacosb+cosasinb cosacosb sinasinb sinacosb cosacosb + cosasinb cosacosb cosacosb cosacosb sinasinb cosacosb??? tana+tanb tana+tanb (abandon ) (definition identities) (ids proven above..) (algebra, divde to get 1 on bottom) (that s why they pay me!!) 6. To prove tan(x y) tanx tany clean up.. 1+tanxtany use the previous one, the tweak; substitute a x and b y.. then pg. 4

5 . Prove the following non-famous identity. cos(4x) cosxcos3x sinxsin3x Solution: ONE way to prove is to start with MOTA, then tweak it.. sub a x and b 3x kachin.. kachin Prove the following non-famous identity. cos(3x) cosxcosx sinxsinx Solution: ONE way to prove is to start with MOTA, then tweak it.. sub a x and b x kachin.. kachin Prove the following non-famous identity. cos(x) cosxcosx sinxsinx Solution: ONE way to prove is to start with MOTA, then tweak it.. sub a x and b x kachin.. kachin Prove the following non-famous identity. cos(4x) cosxcosx sinxsinx Solution: ONE way to prove is to start with MOTA, then tweak it.. sub a x and b x kachin.. kachin Prove the following non-famous identity. cos(5x) cosxcos3x sinxsin3x Solution: ONE way to prove is to start with MOTA, then tweak it.. sub a x and b 3x kachin.. kachin Prove the following non-famous identity. tan(10x) tan1x tanx 1+tan1xtanx Solution: ONE way to prove is to start with the famous id for tan(a b), then tweak it.. sub a 1x and b x kachin.. kachin Prove the following non-famous identity. tan(10x) tan8x+tanx 1 tan8xtanx pg. 5

6 Solution: ONE way to prove is to start with the famous id for tan(a+b), then tweak it.. sub a 8x and b x kachin.. kachin Prove the following non-famous identity. cos(8x) cos10xcosx sin10xsinx Solution: ONE way to prove is to start with MOTA, then tweak it.. sub a 10x and b x kachin.. kachin Prove the following non-famous identity. cos(8x) cos4xcos4x sin4xsin4x Solution: ONE way to prove is to start with MOTA, then tweak it.. sub a 4x and b 4x kachin.. kachin Prove the following non-famous identity. cos10xcosx sin10xsinx cos4xcos4x sin4xsin4x Solution: hint.. look at the previous two problems Without calculators determine if the following is true, then explain... cos Solution: ONE way to prove is to start with MOTA, then tweak it.. sub a 45 and b 30 kachin.. kachin Without calculators determine if the following is true, then explain Solution: I leave some of the fun one for you to try Prove and OWN everyone of these famous identities. Double-Angle Identities cos(a) cos a sin a cos(a) 1 sin a tan(a) tana 1 tan a cos θ 1+cosθ cos(a) cos a 1 sin(a) sinacosa sin θ 1 cosθ pg. 6

7 Solution: try try and they try..if no luck.. then wait we will tackle precisely these in the next few sections Prove and OWN everyone of these famous identities. Product-to-Sums Identities sinasinb 1 [cos(a b) cos(a+b)] cosacosb 1 [cos(a b)+cos(a+b)] sinacosb 1 [sin(a+b)+sin(a b)] Solution: try try and they try..if no luck.. then wait we will tackle precisely these in the next few sections Prove and OWN everyone of these famous identities. Sums-to-Products Identities ( ) ( ) a+b a b sina+sinb sin cos ( ) ( ) a b a+b sina sinb sin cos ( ) ( ) a+b a b cosa+cosb cos cos ( ) ( ) a+b a b cosa cosb sin sin Solution: try try and they try..if no luck.. then wait we will tackle precisely these in the next few sections (**)Prove the identity sin(x) sinxcosx pg. 7

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Trigonometry

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH0000 SEMESTER 1 017/018 DR. ANTHONY BROWN 5. Trigonometry 5.1. Parity and Co-Function Identities. In Section 4.6 of Chapter 4 we looked