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.

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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…

As a matter of fact, right triangles end up being more of a rarity than commonplace. 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 2 Laws of Trigonometry that allow one to solve problems that involve non-right Triangles:

A triangle is uniquely determined by two angles and a particular side A C B c b a O1O1 O2O2

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

Find the length of b A C B c b 5 30 o 65 o

Construct CN with height h A C B c b 5 30 o 65 o N h

By the right triangle SIN ratio A C B c b 5 30 o 65 o N h Sin 30 o = h b Sin 65 o = h 5 h 30 o 65 o

Solve both equations for h X b Sin 30 o = h b Sin 65 o = h 5 X 5 bSin30 o = h h = 5Sin65 o Because the equations are equal bSin30 o = 5Sin65 o

b = 5Sin65 o Sin30 o b = 9.1 Consider the general case:

A C c b a N h B Sin A = h b Sin B = h a bSinA = aSinB

a a bSinA = SinB a b ab b SinA = SinB a

Extend this to all 3 sides of a triangle, and the Sine Law is generated! b SinA = SinB a c = SinC

Find the length of a a A C c 24 73 o 57 o N a Sin73 o =24 Sin57 o a = 27.4

5.9 O 10.3 O 2.9 km Find h h

5.9 O 10.3 O 2.9 km Find h 1. Find O O O = 180 O – 5.9 O – 10.3 O = 163.8 O

5.9 O 10.3 O 2.9 km 163.8 O Find X X X SIN 10.3 O 2.9 SIN163.8 O = X = 1.86km

5.9 O 10.3 O 2.9 km Find h h 1.86 km SIN 5.9 O = h 1.86 km h = 191.2 m

The Ambiguous Case

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

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

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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