Presentation on theme: "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."— Presentation transcript:
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.
There are many situations where angles other than 90 O are present.
Does that mean when we come across a situation that can only be modeled with a non-right triangle that we abandon our pursuit?….
No Way!!!! There exists two Laws of Trigonometry that allow one to solve problems that involve non- right Triangles:
Remember A capital letter represents an angle in a triangle, and a small letter represents a side of a triangle A a
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..)
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
In General: a 2 = b 2 + c 2 – 2bcCosO o A B O C b c a
For Example: Find a A 50 o C 8m 10m a B
a 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
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.
If a corresponding angle and side are known, they form an “opposing pair” A C B c b a O1O1 O2O2
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
The Sine Law b SinA = SinB a c = SinC A C B c b a
Find the length of a a A C c 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)
Find the length of a a A C c o N a Sin73 o =24 Sin57 o a = o Again, this can be put directly into your calculator. See me for help.