Question 813248
Is there a triangle that satisfies a=9 , b=3, and alpha=54 degrees? How do I answer this question? Thanks!
<pre>
By using the law of sines:

{{{a/sin(alpha)}}}{{{""=""}}}{{{b/sin(beta)}}}{{{""=""}}}{{{c/sin(gamma)}}}

Using the first two expressions:

{{{a/sin(alpha)}}}{{{""=""}}}{{{b/sin(beta)}}}

{{{9/sin("54°")}}}{{{""=""}}}{{{3/sin(beta)}}}

Cross-multiply:

9·sin(<font face="symbol">b</font>) = 3·sin(54°)

divide both sides by 9

sin(<font face="symbol">b</font>) = {{{3/9}}}·sin(54°)

sin(<font face="symbol">b</font>) = {{{1/3}}}·sin(54°)

sin(<font face="symbol">b</font>) = .2696723315

So <font face="symbol">b</font> 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 <font face="symbol">g</font> is 180°-(54°+15.645°) = 154.354°.

But, with  <font face="symbol">a</font> = 54° and <font face="symbol">g</font> = 164.354 the sum of the angles would exceed 180º.
Not possible!!!!

Therefore,  <font face="symbol">g</font> = 180°-(54°+15.645°) = 110.355°

and only ONE triangle is possible.

So the one possibility looks like this:

{{{drawing(400,3120/23,-.5,11,-.5,3.4, locate(0,0,beta),locate(10.4,0,alpha),
locate(8.4,3,gamma), locate(9.6,1.6,3), locate(4.5,1.7,9),
locate(9.2,.5,"54°"),
triangle(0,0,10.42992506,0,10.42992506-1.763355757,2.427050983) )}}}

Edwin</pre>