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Published byEdith Twining Modified over 3 years ago

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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…

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

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

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

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No Way!!!! There exists two Laws of Trigonometry that allow one to solve problems that involve non- right Triangles:

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Remember A capital letter represents an angle in a triangle, and a small letter represents a side of a triangle A a

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

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

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In General: a 2 = b 2 + c 2 – 2bcCosO o A B O C b c a

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

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a 2 = 8 2 + 10 2 – 2(8)(10)Cos50 o A 50 o C 8m 10m a a 2 = 61.15m a = 7.8 m B a 2 = b 2 + c 2 – 2(b)(c)CosA o You should be able to load this into your calculator directly from left to right…if not, see me

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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.

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If a corresponding angle and side are known, they form an “opposing pair” A C B c b a O1O1 O2O2

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The Sine Law can be used to determine an unknown side or angle given an “opposing pair” A C B c b a O1O1 O2O2

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

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

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Find the length of a a A C c 24 57 o N a Sin73 o =24 Sin57 o a = 27.4 73 o Again, this can be put directly into your calculator. See me for help.

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

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The Ambiguous Case

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Find A 11 9 48 o A 11 SinA = 9 Sin48 o A=65.3 o Does that make sense? No Way!!!

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

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

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6.1 Law of Sines Objectives –Use the Law of Sines to solve oblique triangles –Use the Law of Sines to solve, is possible, the triangle or triangles in.

6.1 Law of Sines Objectives –Use the Law of Sines to solve oblique triangles –Use the Law of Sines to solve, is possible, the triangle or triangles in.

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