SOLUTION: Two side and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. a=7,b=

There is always exactly one solution when the given 
side that is opposite the given angle is longer than 
the other given side.

In this problem, the given side b is opposite given 
angle B and b is shorter than the other given side a, 
so we cannot tell immediately.

To find out we use the law of sines. If the sine of the 
unknown angle opposite the other given side turns
out to be greater than 1, there is no solution.  If it is less
than 1, there are two solutions, and in very rare cases when it
comes out exactly equal to 1, there is one solution, and it is
a right triangle.   

Using the law of sines:





Cross multiply:







Since this sine came out less than 1, 
there are 2 solutions.

There are two angles with this sine, 

one is

A = 28.60889789°

and the other is its supplement

A = 180° - 28.60889789° = 151.3911021°

Here's the triangle with A = 28.60889789°



We'll solve it first.

We find angle C by adding A and B and subtracting from
180°:

angle C = 180° - (20° + 28.60889789°) = 131.3911021°

We find side c by the law of sines:



Cross multiply:









Here's the triangle with A = 151.3911021°

  

Now we'll solve it:

We find angle C by adding A and B and subtracting from
180°:

angle C = 180° - (20° + 151.3911021°) = 8.608897887°

We find side c by the law of sines:



Cross multiply:









Edwin