SOLUTION: Sketch a picture and solve the triangle (find all side lengths and angle measures to the nearest tenth) using Law of Cosines and/or Law of Sines. Show all your work. (hint: t

Algebra ->  Trigonometry-basics -> SOLUTION: Sketch a picture and solve the triangle (find all side lengths and angle measures to the nearest tenth) using Law of Cosines and/or Law of Sines. Show all your work. (hint: t      Log On


   

Question 1037821: Sketch a picture and solve the triangle (find all side
lengths and angle measures to the nearest tenth) using Law
of Cosines and/or Law of Sines. Show all your work. (hint:
this is an SSA scenario with two solutions, so you’ll need
to sketch and solve two triangles)
∆ABC with m∠A = 38 and AB = 6.3 and BC = 4.2
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
∆ABC with m∠A = 38° and AB = 6.3 and BC = 4.2

Using law of sines:



matrix%281%2C3%2CBC%2Cor%2C%22BC%27%22%29%2Fsin%28%22%3CA%22%29%22%22=%22%22AB%2Fsin%28matrix%281%2C3%2C%22%3CC%22%2Cor%2C%22%3CAC%27B%22%29%29

4.2%2Fsin%28%2238%B0%22%29%22%22=%22%226.3%2Fsin%28matrix%281%2C3%2C%22%3CC%22%2Cor%2C%22%3CAC%27B%22%29%29

4.2sin%28matrix%281%2C3%2C%22%3CC%22%2Cor%2C%22%3CAC%27B%22%29%29%22%22=%22%226.3sin%28%2238%B0%22%29

sin%28matrix%281%2C3%2C%22%3CC%22%2Cor%2C%22%3CAC%27B%22%29%29%22%22=%22%226.3sin%28%2238%B0%22%29%2F4.2

sin%28matrix%281%2C3%2C%22%3CC%22%2Cor%2C%22%3CAC%27B%22%29%29%22%22=%22%220.923492213

∠C is an acute (QI) angle and ∠AC'B is an 
obtuse (QII) angle.  They have the same sine and are
supplementary.

Using the inverse sine feature of our calculator,

∠C = 67.44208077°

so ∠AC'B = 180°-67.44208077° = 112.5579192°

To find side AC and AC', we use the law of cosines:

CB%5E2%22%22=%22%22AC%5E2%2BAB%5E2-2%2AAB%2AAC%2Acos%28%22%3CA%22%29

Swap sides:

AC%5E2%2BAB%5E2-2%2AAB%2AAC%2Acos%28%22%3CA%22%29%22%22=%22%22CB%5E2

Rearrange terms and a factor with 0 on the right:

AC%5E2-2%2AAB%2Acos%28%22%3CA%22%29%2AAC%2BAB%5E2-CB%5E2%22%22=%22%220

AC%5E2-2%2A6.3%2Acos%28%22%3CA%22%29%2AAC%2B6.3%5E2-4.2%5E2%22%22=%22%220

AC%5E2-12.6%2Acos%28%2238%B0%22%29%2AAC%2B22.05%22%22=%22%220

Use the quadratic formula

AC%22%22=%22%22

AC%22%22=%22%22

Using the + gives us AC = 6.575659859
Using the - gives us AC' = 3.353275637

We can find ∠ABC and ∠ABC' by adding the 
two angles we now know and subtracting from 180°.

You can finish that part.

Edwin