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FACTORIZATION &, DEFACTORIZATION, , 64 | Defactorization, , Defactorization formula, , x, (Conversion of product into sum or difference), , , , , , , , , , , » sin A- cos B= sin (A +B) + sin (A~B), B= sin (A +B)-sin (A-B), B= cos (A+ B) + cos (A- B), , Example 6.1: Prove that SF, , \AJ2 008 A- sin, 3) 2 cos A: cos, A. sin B= cos (A-B) ~ cos (A * B), , , , 4) 2 sin, Solution :, !) By using compound angle formulae, we have, \ sin (A +B) = sin A- cos B + cos A sin B wl), sin (AB) = sin A: cos B~ cos A sin B .(2), , Now adding equation (1) and (2), we get

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6-2 Factorization & dep,, , , , , Basic Mathematics, , , , , , = sin A: cos B+ cos A - sj, , sin (A+B) on subtracting equation (1) and (2), we get, +sin(A-B) = sin A cos B = 08 A sing cos (A+B) = cos A. p08 B—sin A. sin B, ot ip sn Al, , cin (A By PMH (A. BY “2 sit A “con cos (A~B) = oo6A - cos B+ sin A sin B, , , , A-cos B = sin (A+B) +sin (A-B), , angle formulae, we have, , n 2sin cos (A+B) -cos (A-B) = ~2sinA-sinB, 2) By using compound Cea ens, , sin (A+B) = sin A- cos B+cos A- sinB, , cos (A, , 2 sin A - sin B = cos (A — B) — cos (A+B), , , , sin (A-B) = sin A- cos B— cos A - sinB, , +) ; 6.2: Express the following as sum or difference of trigonometri, Now subtracting equation (1) and (2), we get metric, sin(A+B) = sin A-cosB+cos A-sinB 7 2) 2 sin 60° cos 20°, sin(A~B) ~ on A oon 5 Ae nO )sin 11°A sin A 4) 2 cos 75° cos 15°, rere eee ae, sin(A+B)-sin(A—B) = 2cosA-sinB ‘Summer 2014, 5) 00s “7 COS "7 6) sin 17° - sin 13, , a 2.cos A: sin B = sin (A + B) — sin (A — B), 7) sin (135° -A) sin (135° +A) 8) cos (120° - A) cos (120° + A), , 9) sin (60° - A) - sin (60° + A) 10) 4 cos 30° - sin 20°, , Solution :, , 1) 2sin 70° cos 30°, , 3) By using compound angle formulae, we have, cos (A +B) = cos A: cos B— sin A: sinB, cos (A—B)=cosA+cosB+sin A: sinB, Now add equation (1) and (2), we get, cos (A+B) = cos A- cos B— sin A - sii, +cos(A-B) = cos A- cos B + si “sin B, , mo ee, , cos (A+B) +cos(A-B) = 2cosA-cosB, , , , , , , , i, , , , , , , sin (A + B) + sin (A-B), , , , 2 sin A cos B, , , , , , sin (70° + 30°) + sin (70° — 30°), = sin (100°) +sin (40°), , , , 2.cos A - cos B = cos (A + B) + cos (A—B), 4) By using compound angle formulae, we have, cos (A +B) = cos A- cosB-sin A- sinB lly, cos (A~B) = cosA-cosB+sin A: sinB fd), ene at, re, , Technical Pt mu, 1S - An up thrust for knowledge, , , , , , 2 sin 70° cos 30° = sin 100° + sin 40°, , , , , , 2)2 sin 60° cos 20°, , , , , 2 sin A-cos B= sin (A+B) +sin (A-B), , , , Site ee, , Anup thrust for ‘knowledge, , ™, T Technical Publications

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6-4 Factorization & p, ef, , Clary,, sin (60° + 20°) + sin (60° 205), sin 80° + sin 40°, , Basic Mathematics, , , , : é, mathematics 5 atrction &Defacon, 2.008 A cos B= 00s (A +B) + ogg, , , , , , , , , , E: 2 sin 60° cos 20° = sin 80° + sin 40° es 3 {os(, 3)sin11AsinA 1 { (@ E 2), = 5)cos + cos |=, . 7 ay, Multiply and divide by 2 ; : 7 Cos(— 8) = cos @, 2sin 1A -sinA _1 (2)- a), = 2 =o {eos 7 cos 7, 3m) 1, 2 sin A- sin B = cos (AB) - cos (A+B) cde (72)- cos 3) = jem ($) <n (4), , , , 6)sin 17° - sin 13°, Multiply and divide by 2, 1, = 2 {2 sin 17°- sin 13°), , , , 1, : sin 11 A sin A = 5 [cos 10 A ~cos 12 A], , , , 2 sin A-sinB = cos (A ~ B) ~ cos (A +B), , 4) 2 cos 75° cos 15° 1, =3 {cos (17- 13) - cos (17 + 13), , = cos (75° + 15°) + cos (75° - 15°) ay ; {cos (4)- cos 30} |cos 30° 8, , , , , , , , MI, , , , = cos (90°) + cos (60°), oe at E 3, O45 =5 =} foo 4) --F} cos 4) +38, D 1, “ 7 €08 75°. cos In=> sin 17° - sin 13° = 1054+, 2n, 5) cos ( 2) Sie ( sn) 7)sin (135° — a) sin (138° + A), , | Multiply and divide by2, = i {2 sin (135° — A) sin (135° + A)}, , 2 sin A - sin B = cos (A- B)- cos (A + B), Se tains” anponaein = ———, , ™, Technical Publications, , Multiply and divide by 2

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Basic Mathematics, , , , Factorization & Defacy, re,, , = ze (135°- A- 135°) Oe — ebay, , = 5 {eos (2A) - cos (270°)}, , = i {cos 2A - 0}, , 1, , , , = 3008 2A, sin (135° - A) sin (135° + A)= 2 1 nos 2A, 8) 4cos 30°- sin 20° <Sumimer ag, = 2(2 cos 30° sin 20°), , , , 2cos A sin B = sin (A + B) — sin (A-B), , , , , , , , = 2{ sin (30° + 20°) — sin (30° - 20%),, = 2{ sin (50°) — sin 10°}, 4 cos 30° sin 20° = 2 [sin 50° — sin 10°], , mple 6.3 : If 2 sin 50° - cos 70° =, , sin A-sinB, fine fend B., , , , , , , , Solution: Given : 2 sin 50° - cos 70° = sin A - sin B ll, On L.HLS. by defactorization formula, we have, 2 sin A - cos B = sin(A + B) + sin (A-B), Equation (1) becomes, , sin (50° + 70°) + sin (50° — 70°) = sin A- sin B, , sin 120° + sin (- 20°) = sin A-sinB, , a iin ti ch A ge, , T, , sin 120° — sin 20° = sin A— sin B, , ™, Technical Publications - An up thrust for knowledge, , , , , , , , , , , , mathematics Factorization & Defactorization, of equating the term on both sides, joke sin 120° and sinB = sin 20°, A = 120° “~~ B=29°, , , , golution : Given : 2 cos 60° - cos 10° = cos A+ cos B £2), on L.LS. by defactorization formula, we have, , , , , 2 cos A- cos B= cos (A+ B)+cos (A -B), 2, cos 60° - cos 10° = cos (60 + 10) + cos (60 — 10), Fquation (1) becomes, cos (60+ 10) + cos (60 — 10) = cos A+ cos B, cos (70) + cos (50) = cos A + cos B, On equating the terms on both sides, , cos A = cos 70°, A = 70° &, , and cos B = cos (50), , B =50°, , , , Example 6.5: If 2 sin 40° cos 10° =sinA+sinB, then inet Rane B, , 5 (MSBTES, , (Q), , Solution: Given :, On LHS. by defactorization formula, +, 2 sin A- cos B= sin (A+B) +sin(A- B), 2 sin 40° - cos 10° = sin (40 + 10) + sin (40 - 10), Equation (1) becomes, sin (40 + 10) + sin (40 - 10) = sinA+sinB, sin 50 + sin 30 = sinA* sin B, , 2 sin 40° « cos 10° =sin A+ sinB, have, , , , |, , , , T Technical Publications” ~ An up thrust for knowledge

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ES MSBTE: 199;, , , , 3m It, solution: Given sin“ -sin7D — cos 51+ cos 67%, Multiply and divide by 2, tf 3. gin 2-2 008 Secs, =3/28i2* sin“> cos £086, , , , , = cos (A- B)— cos (A+B), ) + cos (A-B), , , , 2sin A-sinB, , 2 cos A -cos B= cos (A+B, 7 3:, , [om (22- 18) cos (F+ 1B) (eos (Sn + 6n) + 008 (m- «n)|, , 1 =, | 0089 cos, , 2m) - cos Sm — (cos 11m + cos (-7))], , , , i, , i], , 1, 2 [cos (, cos (— 8) = 0084, , el, = 7 [cos 2n- cos St — (cos 11% + cos 7)], , i, = 7 [cos 2 — cos Sm — cos 11n-cos 7] (costo =C 5), , 1, =g0-CD-C)-CH), , 1, = 5 [+141 +1]=$=, , T Technic, a! Publications - An up thrust for knowledge, , , , , , Factorization & Defactorization, , , , sim 5°, a4, , , , , , , = sin (A+ B)+ sin (A-B), = sin (A+B)-sin (A—B), 5° + 5°) + sin (25° - 5°), , 70°), , L fsin@, — (gin (80° + 70°) — - sin (80° — (sin 150° - sin 10°), , pa 3 [sin 30° + sin 20°, , = 5 [sin 30° + sin 20° — sin 150° + sin 10°], , 2 $Isin 30° + sin 20° — sin (180° ~ 30°) + sin 10°], , , , sin 30° + sin 10°], , = “aie 30° + sin 20° —, , 5 [sin 20° + sin 10°]