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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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9-2 Sine and Cosine Ratios. There are two more ratios in trigonometry that are very useful when determining the length of a side or the measure of an.

9-2 Sine and Cosine Ratios. There are two more ratios in trigonometry that are very useful when determining the length of a side or the measure of an.

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