SOLUTION: Determine if the following triangles have no solutions, one solution, or two solutions: 1. A=130 degrees, a=19, b=11. 2. A=45 degrees, a=4√2, b=8. 3. A=32 degrees, a=

Algebra ->  Trigonometry-basics -> SOLUTION: Determine if the following triangles have no solutions, one solution, or two solutions: 1. A=130 degrees, a=19, b=11. 2. A=45 degrees, a=4√2, b=8. 3. A=32 degrees, a=      Log On


   

Question 42529: Determine if the following triangles have no solutions, one solution, or two solutions:
1. A=130 degrees, a=19, b=11.
2. A=45 degrees, a=4√2, b=8.
3. A=32 degrees, a=16, b=21.
4. A=90 degrees, a=25, b=15.
Thank You
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
I am showing the procedure to solve this type of problems. I am solving the first question. Do the rest yourself. That will nake you understand this things better.
For the first problem, A=130%5Eo, a=19, b=11.

The sine rule states:
a%2Fsin%28A%29+=+b%2Fsin%28B%29

19%2Fsin%28130%5Eo%29+=+11%2Fsin%28B%29
or 19%2Asin%28B%29+=+11%2Asin%28130%5Eo%29
or sin%28B%29+=+%2811%2Asin%28130%5Eo%29%29%2F19
or sin%28B%29+=+%2811%2A0.766%29%2F19
or sin%28B%29+=+0.4435
or B+=+sin%5E-1%280.4435%29
or B+=+26.33%5Eo

Hence, C+=+180%5Eo+-+A+-+B
or C+=+180%5Eo+-+135%5Eo+-+26.33%5Eo+=+18.67%5Eo

Now, apply sine rule again
a%2Fsin%28A%29+=+c%2Fsin%28C%29
or 19%2Fsin%28135%5Eo%29+=+c%2Fsin%2818.67%5Eo%29
or c+=+%2819%2Asin%2818.67%5Eo%29%29%2Fsin%28135%5Eo%29
or c+=+%2819%2A0.32%29%2F0.766
or c+=+7.94

Now, we observe a > b + c [since 19 > 11 + 7.94].
But, for a triangle to exist, the sum of any two sides must be greater than the third.
So this triangle does not exist.
So there is no solution.


Solve the other problems in a similar way.
If the condition: "Sum of any two sides of a triangle is greater than the third" is not violated then the triangle exists and hence the solution is unique i.e. only one solution. Otherwise there is no solution.