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TRIG IDENTITIES (II)

TRIG IDENTITIES (II). The following relationships are always true for two angles A and B. (1)(a) sin(A + B) = sinAcosB + cosAsinB. Supplied on a formula sheet !!. (1)(b) sin(A - B) = sinAcosB - cosAsinB. (2)(a) cos(A + B) = cosAcosB - sinAsinB. (2)(a) cos(A - B) = cosAcosB + sinAsinB.

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TRIG IDENTITIES (II)

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  1. TRIG IDENTITIES (II) The following relationships are always true for two angles A and B. (1)(a) sin(A + B) = sinAcosB + cosAsinB Supplied on a formula sheet !! (1)(b) sin(A - B) = sinAcosB - cosAsinB (2)(a) cos(A + B) = cosAcosB - sinAsinB (2)(a) cos(A - B) = cosAcosB + sinAsinB Quite tricky to prove but some of following examples should show that they do work!!

  2. Examples 1 (1) Expand cos(U – V). (use formula (2)(b) ) cos(U – V) = cosUcosV + sinUsinV (2) Simplify sinf°cosg° - cosf°sing° (use formula (1)(b) ) sinf°cosg° - cosf°sing° = sin(f – g)° (3) Simplify cos8sin + sin8cos (use formula (1)(a) ) cos8sin + sin8cos = sin(8 + ) = sin9

  3. Example 2 By taking A = 60° and B = 30°, prove the identity for cos(A – B). ***************** NB: cos(A – B) = cosAcosB + sinAsinB If A = 60° and B = 30° then LHS = cos(60 – 30 )° = cos30° = 3/2 RHS = cos60°cos30° + sin60°sin30° = ( ½ X 3/2 ) + (3/2 X ½) = 3/4 + 3/4 = 3/2 Hence LHS = RHS !!

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