1 / 11

Aim: How do we find the cosine of the difference and sum of two angles?

Aim: How do we find the cosine of the difference and sum of two angles?. If m A = 35º and m B = 20º show which of the following identities is true?. Do Now:. cos(A – B) = cos A – cos B. cos(35º – 20º) = cos 35º – cos 20º. cos(15º) = .8191520443 – .9396926208. .9659258263 = -.1205405765.

hanne
Télécharger la présentation

Aim: How do we find the cosine of the difference and sum of two angles?

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. Aim: How do we find the cosine of the difference and sum of two angles? If mA = 35º and mB = 20º show which of the following identities is true? Do Now: cos(A – B) = cos A – cos B cos(35º – 20º) = cos 35º – cos 20º cos(15º) = .8191520443 – .9396926208 .9659258263 = -.1205405765 cos(A – B) = cosAcosB +sinASinB .9659258263 = cos 35•cos 20 + sin 35•sin20 .9659258263 = .9659258263

  2. P(cos B, sin B) Q(cos A, sin A) (x,y) A – B B A The Cosine of the Difference of 2 Angles cos (A – B) = cos A cos B + sin A sin B Outline of Proof – Cos(A – B) y 1 O x -1 1 Find the length of PQ using both the Law of Cosines and the distance formula, equating the two to arrive at:

  3. Model Problem Use the identity for the cosine of the difference of two angle measures to prove that cos (180º – x) = -cos x. cos (A – B) = cos A cos B + sin A sin B Substitute 180 for A and x for B: cos (180 – x) = cos 180 cos x + sin 180 sin x simplify: cos (180 – x) = -1 • cos x + 0 • sin x cos (180 – x) = -cos x

  4. Model Problem If sin A = 3/5 and A is in QII, and cos B = 5/13 and B is in QI, find cos (A – B). cos (A – B) = cos A cos B + sin A sin B to use this you need to know both sine and cosine values for both A and B. HOW? Pythagorean Identity sin2 A + cos2 A = 1 sin2 B + cos2 B = 1 sin2 B + (5/13)2 = 1 (3/5)2 + cos2 A = 1 9/25 + cos2 A = 1 sin2 B + 25/169 = 1 sin2 B = 144/169 cos2 A = 16/25 sin B = 12/13 cosA = 4/5 QII cosA = -4/5 QI

  5. A: sin A = 3/5, cosA = -4/5 B: cos B = 5/13, sin B = 12/13 Model Problem (con’t) If sin A = 3/5 and A is in QII, and cos B = 5/13 and B is in QI, find cos (A – B). cos (A – B) = cos A cos B + sin A sin B Substitute & simplify: cos (A – B) = (-4/5)(5/13) + (3/5)(12/13) cos (A – B) = (-20/65) + (36/65) cos (A – B) = (16/65)

  6. cos (-x) = cos (x) sin (-x) = -sin x The Cosine of the Sum of 2 Angles: cos (A + B) = cos A cos B – sin A sin B Cosine of Sum of 2 Angles Prove: cos (A + B) = cos A cos B – sin A sin B cos (A + B) = cos (A - (-B)) cos (A + B) = cos A cos(-B) + sin A sin(-B) cos (A + B) = cos A(cos B) + sin A(-sinB) cos (A + B) = cos A cos B – sin A sinB ex. Show that cos 90 = 0 by using cos(60 + 30) cos (60 + 30) = cos 60 cos 30 – sin 60 sin 30 = 0

  7. The Cosine of the Sum of 2 Angles: cos (A + B) = cos A cos B – sin A sin B Model Problem Find the exact value of cos45ºcos15º – sin45ºsin15º cos(45 + 15) =cos45ºcos15º – sin45ºsin15º cos(60) = 1/2 Find the exact value of cos 75 by using cos (45 + 30) cos (45 + 30) = cos 45 cos 30 – sin 45 sin 30

  8. More Proofs Prove: cos (-x) = cos x cos (A – B) = cos A cos B + sin A sin B cos (0 – x) = cos 0 cos (x) + sin 0 sin (x) cos (0 – x) = 1 • cos (-x) + 0 • sin (-x) cos (-x) = cos (x) Prove: sin (-x) = -sin x the sine of an angle equals the cosine of its complement sin  = cos(90 – ) , let  = -x sin (-x) = cos(90 – (-x)) = = cos (90 + x) = cos (x + 90) = cos (x – (-90) = cos x cos (-90) + sin x sin (-90) = cos x • 0 + sin x • -1 = -sin x

  9. Model Problem

  10. Regents Question

  11. The Product Rule

More Related