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Lesson 35 – The Cosine Law

Lesson 35 – The Cosine Law. Integrated Math 10 – Santowski. Cosine function for triangle ADB cos A = c/x  x = c cos A Pythagorean theorem for triangle ADB x 2 + h 2 = c 2  h 2 = c 2 − x 2 Pythagorean theorem for triangle CDB ( b − x ) 2 + h 2 = a 2. (A) Cosine Law - Derivation.

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Lesson 35 – The Cosine Law

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  1. Lesson 35 – The Cosine Law Integrated Math 10 – Santowski Integrated Math 10 - Santowski

  2. Cosine function for triangle ADBcosA=c/x x=ccosA Pythagorean theorem for triangle ADBx2+h2=c2 h2=c2−x2 Pythagorean theorem for triangle CDB(b−x)2+h2=a2 (A) Cosine Law - Derivation Integrated Math 10 - Santowski

  3. (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)=a2b2−2bx+c2=a2 • Substitute x = c cos Ab2−2b(ccosA)+c2=a2 • Rearrange: • a2=b2+c2−2bccosA Integrated Math 10 - Santowski

  4. (C) Law of Cosines: A b Have: two sides, included angle Solve for: missing side c C B a c2 = a2 + b2 – 2 ab cos C (missingside)2= (one side)2+ (otherside)2 – 2(one side)(other side) cos(includedangle) Integrated Math 10 - Santowski

  5. (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 = Integrated Math 10 - Santowski

  6. B c=5.2 a=2.4 A b=3.5 C (D) Cosine Law - Examples • Solve this triangle Integrated Math 10 - Santowski

  7. B c=5.2 a=2.4 A b=3.5 C (D) Cosine Law - Examples 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. Integrated Math 10 - Santowski

  8. (D) Cosine Law - Examples Integrated Math 10 - Santowski

  9. (D) Cosine Law - Examples Integrated Math 10 - Santowski

  10. (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 Integrated Math 10 - Santowski

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