SOLUTION: The following facts are true about triangle ABC: [measure_angle_B=120 degrees, AC=x(sqroot3), BC=x How many choices are there for the measure of angle A?

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Question 205114: The following facts are true about triangle ABC:
[measure_angle_B=120 degrees, AC=x(sqroot3), BC=x
How many choices are there for the measure of angle A?
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
The following facts are true about triangle ABC:
[measure_angle_B=120 degrees,
AC=x(sqroot3) (this is side "b",
BC=x (this is side a)
How many choices are there for the measure of angle A?
----
Using the Law of Sines you get:
sin(A)/a = sin(B)/b
-------------------
sin(A)/x = sin(120)/x(sqrt(3))
---
sin(A) = [sqrt(3)/2]/(sqrt(3))
sin(A) = (1/2)
---
So the measure of angle A could be 30 deg or it could be 150 deg
=====================================================================
Cheers,
Stan H.

Answer by Edwin McCravy(20086) About Me  (Show Source):
You can put this solution on YOUR website!
Edwin's solution. Stanbon is wrong to say angle A could be 150°. It can only be 30°. Here's why: 


By the law of sines:

BC%2Fsin%28A%29=AC%2Fsin%28B%29


x%2Fsin%28A%29=%28x%2Asqrt%283%29%29%2Fsin%28%22120%B0%22%29 

Cross-multiply:

x%2Asqrt%283%29sin%28A%29=x%2Asin%28%22120%B0%22%29

Divide both sides by x%2Asqrt%283%29





sin%28A%29=sin%28%22120%B0%22%29%2Fsqrt%283%29

We know that sin%28%22120%B0%22%29=sqrt%283%29%2F2, so substituting:

sin%28A%29=%28sqrt%283%29%2F2%29%2Fsqrt%283%29

Write as a division:

sin%28A%29=%28sqrt%283%29%2F2%29%22%F7%22sqrt%283%29

Invert and multiply

sin%28A%29=%28sqrt%283%29%2F2%29%22%2A%22%281%2Fsqrt%283%29%29

sin%28A%29=%28cross%28sqrt%283%29%29%2F2%29%22%2A%22%281%2Fcross%28sqrt%283%29%29%29

sin%28A%29=1%2F2

There are two angles between 0° and 180° which
have sine 1%2F2, they are 30° and 150°, but

angle A cannot be 150°, because angle B is 120°,
and that would make two angles in the same
triangle with sum more than 180° which is 
impossible.  So angle A can only be 30°.

Edwin