SOLUTION: Is there a triangle that satisfies a=9 , b=3, and alpha=54 degrees? How do I answer this question? Thanks!

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Question 813248: Is there a triangle that satisfies a=9 , b=3, and alpha=54 degrees? How do I answer this question? Thanks!
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
Is there a triangle that satisfies a=9 , b=3, and alpha=54 degrees? How do I answer this question? Thanks!
By using the law of sines:

a%2Fsin%28alpha%29%22%22=%22%22b%2Fsin%28beta%29%22%22=%22%22c%2Fsin%28gamma%29

Using the first two expressions:

a%2Fsin%28alpha%29%22%22=%22%22b%2Fsin%28beta%29

9%2Fsin%28%2254%B0%22%29%22%22=%22%223%2Fsin%28beta%29

Cross-multiply:

9·sin(b) = 3·sin(54°)

divide both sides by 9

sin(b) = 3%2F9·sin(54°)

sin(b) = 1%2F3·sin(54°)

sin(b) = .2696723315

So b 15.645° - in Quadrant I.

Sine is also positive in Quadrant II.  If we use the reference 
angle 15.646º in Quadrant II, the angle g is 180°-(54°+15.645°) = 154.354°.

But, with  a = 54° and g = 164.354 the sum of the angles would exceed 180º.
Not possible!!!!

Therefore,  g = 180°-(54°+15.645°) = 110.355°

and only ONE triangle is possible.

So the one possibility looks like this:



Edwin