Integrated Math 10 – Santowski

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Integrated Math 10 – Santowski Lesson 35 – The Cosine Law Integrated Math 10 – Santowski 9/16/2019 Integrated Math 10 - Santowski

(A) Cosine Law - Derivation Cosine function for triangle ADB cosA=c/x  x=ccosA Pythagorean theorem for triangle ADB x2+h2=c2  h2=c2−x2 Pythagorean theorem for triangle CDB (b−x)2+h2=a2 9/16/2019 Integrated Math 10 - Santowski

(A) Cosine Law - Derivation Pythagorean theorem for triangle CDB (b−x)2+h2=a2 Substitute h2 = c2 - x2 (b−x)2+(c2−x2)=a2 (b2−2bx+x2)+(c2−x2)=a2 b2−2bx+c2=a2 Substitute x = c cos A b2−2b(ccosA)+c2=a2 Rearrange: a2=b2+c2−2bccosA 9/16/2019 Integrated Math 10 - Santowski

Integrated Math 10 - Santowski (C) Law of Cosines: A b Have: two sides, included angle Solve for: missing side c C B a c2 = a2 + b2 – 2 a b cos C (missing side)2 = (one side)2 + (other side)2 – 2 (one side)(other side) cos(included angle) 9/16/2019 Integrated Math 10 - Santowski

Integrated Math 10 - Santowski (C) Law of Cosines: A Have: three sides Solve for: missing angle b c C B a Side Opposite Missing Angle Missing Angle a2 + b2 – c2 2ab cos C = 9/16/2019 Integrated Math 10 - Santowski

(D) Cosine Law - Examples Solve this triangle a=2.4 c=5.2 b=3.5 A B C 9/16/2019 Integrated Math 10 - Santowski

(D) Cosine Law - Examples b=3.5 A B C Start with the law of cosines because there are no angles given. a2=b2+c2-2bc cosA. Substitute values. 2.42=3.52+5.22-2(3.5)(5.2) cosA, 5.76-12.25-27.04=-2(3.5)(5.2) cos A, 33.53=36.4cosA, 33.53/36.4=cos A, 0.921=cos A, A=67.07. Now for B. b2=a2+c2-2ac cosB, (3.5)2=(2.4)2+(5.2)2-2(2.4)(5.2) cosB, 12.25=5.76+27.04-24.96 cos B. 12.25=5.76+27.04-24.96 cos B, 12.25-5.76-27.04=-24.96 cos B. 20.54/24.96=cos B. 0.823=cos B. B=34.61. C=180-34.61-67.07=78.32. 9/16/2019 Integrated Math 10 - Santowski

(D) Cosine Law - Examples 9/16/2019 Integrated Math 10 - Santowski

(D) Cosine Law - Examples 9/16/2019 Integrated Math 10 - Santowski

(D) Examples Cosine Law We can use these new trigonometric relationships in solving for unknown sides and angles in acute triangles: ex 1. Find c in CDE if C = 56°, d = 4.7 and e = 8.5 ex 2. Find G in GHJ if h = 5.9, g = 9.2 and j = 8.1 ex 3. Solve CDE if D = 49°, e = 3.7 and c = 5.1 9/16/2019 Integrated Math 10 - Santowski