6.1 Law of Sines If ABC is an oblique triangle with sides a, b, and c, then A B C c b a

Ex. 1Given a triangle with C = o, and B = 28.7 o, and b = 27.4 feet, find the remaining angle and sides. Given: AAS A = B - C A = 49 o A B C c a b = 27.4’ 28.7 o o Set up and find a. Now find side c.

For the SSA oblique triangle there are 3 possible situations that can occur: 1. No triangle exists h > a 2. One triangle exists a > b 3. Two distinct triangles exist. h < a < b Let’s look at the first and third cases.

Case 1:No-solution case - SSA Where h > a Given: a = 15, b= 25, and A = 70 o b = 25 a = 15 A How do we find the height of the triangle? h h = which is longer than opposite side a. In other words, h > a which implies no solution.

Case 2: Two solutions h < a < b Draw two triangles. Given: a = 12 m, b = 31 m, and A = 20.5 o b = 31 B B’ a = 12 c’ c h A C’ C A’

The first thing we need to do after the triangles are drawn is to find the height to see is the triangle even exists. Find the height. h = which is less than 12. Now we can find B, B’, C, C’, c, c’ First, find B.B = 64.8 o Now, find C and C’ C = 94.7 o C’ = 44.3 o Now, use law of sines to find c and c’. c = c’ = 23.93

The Area of an Oblique Triangle A = (1/2) bh Ex. Find the area of the given triangle. 102 o a = 90 m b = 52 m o. Draw in the height and find it by finding the supplement to 102 o. h 78 o h = m