Trignometry Imp Formulae

Trigonometric Formulae July 15, 2009 1 Basic Trigonometry 1. sin2 + cos2 = 1 or sin2 = 1 􀀀 cos2 or sin 1 􀀀 cos = 1 + cos sin 2. sec2 􀀀 tan2 = 1 or sec2 = 1 + tan2 or sec 􀀀 tan = 1 sec + tan 3. cosec2􀀀cot2 = 1 or cosec2= 1+cot2 or cosec 􀀀cot = 1 cosec + cot 2 Sum and Dierence of Angle formulae 1. sin(A B) = sinAcosB cosAsinB 2. cos(A B) = cosAcosB sinAsinB (a) Derived from above : i. sin(A + B) sin(A 􀀀 B) = sin2 A 􀀀 sin2 B = cos2 B 􀀀 cos2 A ii. cos(A + B) cos(A 􀀀 B) = cos2 A 􀀀 sin2 B = cos2 B 􀀀 sin2 A iii. sin 2= 2 sin cos = 2 tan 1 + tan2 iv. cos 2= cos2 􀀀 sin2 = 2 cos2 􀀀 1 = 1 􀀀 2 sin2 = 1 􀀀 tan2 1 + tan2 Another form : cos2 = 1 + cos 22 & sin2 = 1 􀀀 cos 22 v. tan 2= 2 tan 1 􀀀 tan2 3. tan(A B) = tanA tanB 1 tanAtanB (where A;B; (A B) 2 fj2 < 􀀀 fn+ 2 gg) (a) Special Case : tan(4 ) = 1 tan 1 tan 4. cot(A B) = cotAcotB 1 cotA cotB 12 5 PRODUCT TO SUM OR DIFFERENCE 3 Triple Angles 1. sin 3x = 3 sin x 􀀀 4 sin3 x 2. cos 3x = 4 cos3 x 􀀀 3 cos x 3. tan 3x = 3 tan x 􀀀 tan3 x 1 􀀀 3 tan2 x 4 Sum and dierence to Product 1. sinC + sinD = 2 sin C + D 2 cos C + D 2 2. sinC 􀀀 sinD = 2 cos C 􀀀 D 2 sin C + D 2 3. cosC + cosD = 2 cos C + D 2 cos C 􀀀 D 2 4. cosC 􀀀 cosD = 2 sin C + D 2 sin D 􀀀 C 2 (Note D 􀀀 C) 5 Product to sum or dierence 1. 2 sinAcosB = sin(A + B) + sin(A 􀀀 B) 2. 2 cosAsinB = sin(A + B) 􀀀 sin(A 􀀀 B) 3. 2 cosAcosB = cos(A + B) + cos(A 􀀀 B) 4. 2 sinAsinB = cos(A 􀀀 B) 􀀀 cos(A + B) (Note the change here)