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As a matter of fact, right triangles end up being more of a rarity than commonplace.

There is no doubt that the 3 PTRs are extremely useful when solving problems modeled on a right triangle. Unfortunately, the world does not consist only of right triangles…. As a matter of fact, right triangles end up being more of a rarity than commonplace.

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As a matter of fact, right triangles end up being more of a rarity than commonplace.

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  1. There is no doubt that the 3 PTRs are extremely useful when solving problems modeled on a right triangle. Unfortunately, the world does not consist only of right triangles…

  2. As a matter of fact, right triangles end up being more of a rarity than commonplace.

  3. There are many situations where angles other than 90O are present.

  4. Does that mean when we come across a situation that can only be modeled with a non-right triangle that we abandon our pursuit?….

  5. No Way!!!! There exists two Laws of Trigonometry that allow one to solve problems that involve non-right Triangles: The Law of Sines The Law of Cosines

  6. RememberA capital letter represents an angle in a triangle, and a small letter represents a side of a triangle a A

  7. The Law of Cosines

  8. (the blue line also forms a “c”, [kind of] which is how I remember to use the “c”osine law in this case..) If b,c and O are all known, then O is called a “Contained Angle” C a b O B A c

  9. The Cosine Law can be used to find the length of the opposite side to O In this case, the length of side a C a b O B A c

  10. In General: a2 = b2 + c2 – 2bcCosOo C a b O B A c

  11. For Example: Find a C a 8m 50o B A 10m

  12. a2 = b2 + c2 – 2(b)(c)CosAo a2 = 82 + 102 – 2(8)(10)Cos50o a2 = 61.15m You should be able to load this into your calculator directly from left to right…if not, see me a = 7.8 m C a 8m 50o B A 10m

  13. The Law of Sines

  14. The Sine Law If the triangle being solved does not consists of a right triangle (3PTRs) or a contained angle (Cosine Law), then another tool must be used.

  15. If a corresponding angle and side are known, they form an “opposing pair” A C b a O2 O1 B c

  16. The Sine Law can be used to determine an unknown side or angle given an “opposing pair” A C b a O2 O1 B c

  17. The Sine Law SinA = SinB = SinC a b c C a b A B c

  18. Find the length of a We can not use the Cosine Law because there is not a contained angle… We must therefore look for an opposite pair. Hmmm….. C a 57o A-HA!!! (it’s all good) c 73o N A 24

  19. Find the length of a C a = 24 a Sin73o Sin57o 57o a = 27.4 Again, this can be put directly into your calculator. See me for help. c 73o N A 24

  20. Pg 290 1a,c,d,e 4a,c,e 5a,c 6 8,10,12,14 Pg 295 1 (11 unco, stop here)

  21. The Ambiguous Case

  22. Find A 11 SinA Sin48o = 11 9 9 A = 65.3o Does that make sense? 48o A No Way!!!

  23. Side 9 can also be drawn as: Could A be 65o in this case? 11 9 48o A

  24. This type of discrepancy is called the “Ambiguous Case”Be sure to check the diagram to see which answer fits:O, or 180o - O

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