SOLUTION: Two sides of an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangles that result. a= 18 b=

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Question 611875: Two sides of an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangles that result.
a= 18
b= 12
B= 20 degrees
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Just to be clear: The two sides given are not sides of the given angle. They are sides of an angle we do not know, angle C:
  • Lowercase letters are used for sides of a triangle and uppercase letters are used for the angles opposite those sides. Your problem gives you side b and the angle opposite that side, angle B, and side a. Draw a diagram. You'll see that sides a and b are not sides of angle B.
  • If the two sides given were sides of the given angle, then by SAS there would always one possible triangle.
The problem is to find out how many angles could be found between the given sides and still make a triangle. For this we can use either the Law of Sines or the Law of Cosines. Since the Law of Sines is simpler I will use it. Putting the data we're given into the Law of Sines we get:
sin%28A%29%2Fa+=+sin%28B%29%2Fb
sin%28A%29%2F18+=+sin%2820%29%2F12
Multiplying each side by 18 we get:
sin%28A%29+=+18%2Asin%2820%29%2F12
With our calculators we can simplify this to:
sin(A) = 0.51303021

The question now is: How many angles could A if its sin is 0.51303021 and if it has to fit inside a triangle with the sides and angle given.?

We can use the inverse function on our calculator to find the reference angle:
Reference angle for A = 30.86588247 degrees
An angle inside of a triangle has to be less than 180 degrees. Using this and the reference angle we can find that:
A = 30.86588247 degrees
or
A = 180 - 30.86588247 = 149.13411753 degrees.

The only question remaining is: Can both of these angles fit inside of a triangle that already has a 20 degree angle? The answer to this is: Yes. With the first A there would be approximately 150 degrees left for the third angle and for the second A there would be approximately 10 degrees left for the third angle.

So there are two possible triangles with the given information.

For problems like this In general:
  • There will be no triangles if the sin value is impossible, like sin(A) = 1.2 (since sin can never be more than 1)
  • There will be just one triangle if...
    • the given angle is 90 degrees or more; or
    • the reference angle is less than or equal to the given angle. For example, if the given angle is 43 degrees and the reference angle turns out to be 43 or less.
  • There will be two possible triangles, like in this problem, in any other case.