1 / 6

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)(b) cos(A - B) = cosAcosB + sinAsinB.

glenna
Download Presentation

TRIG IDENTITIES (II)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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)(b) cos(A - B) = cosAcosB + sinAsinB Quite tricky to prove but some of following examples should show that they do work!!

  2. Consider (cosA, sinA) Q Q’ A+B A (1,0) (1,0) O O -B (cosB, -sinB) P’ P Rotate ∆OPQ about O through angle B

  3. PQ2 = (cos A – cos B)2 + (sin A + sin B)2 = cos2 A + sin2 A + cos2 B +sin2 B – 2cosAcosB + 2sinAsinB = 2 – 2(cosAcosB – sinAsinB)

  4. P′ is (1,0) and Q′ is (cos(A+B), sin(A+B)) (P′Q′)2 = (1 – cos(A+B))2 + (sin(A+B))2 = 1 + cos2(A+B) + sin2(A+B) – 2cos(A+B) = 2 - 2cos(A+B) BUT PQ2 = (P′Q′)2 So it follows that cos (A+B) = cosAcosB – sinAsinB AND cos (A+(-B))= cos(A-B) = cosAcosB-sinAsinB

  5. 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

  6. 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 !!

More Related