Solve this triangle: a=8.4 b=22.1 A=33°46'
.C
b = 22.1 . \
. \a = 8.4
. \
A. 33°46' \B
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This is the ambiguous case. Sometimes there is no solution,
sometimes there is one solution, and sometimes there are two solutions.
There is no solution when a is too short to reach down to AB.
If the picture above shows one possible solution, here is another:
.C
b = 22.1 . /
. /a = 8.4
. /
A. 33°46' / B
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The side a can slant right or left if it is long enough to reach
down to AB. Sometimes a is so long that it can only slant to the right,
and if we slant it to the left it extends left of A, and not create a
second solution. That would be the case when there would be only one
solution. We don't yet know how many solutions, if any, we have.
In any case we use the law of sines:
a b c
———— = ———— = —————
sinA sinB sinC
We only need this part of the law of sines:
a b
———— = ————
sinA sinB
We solve for sinB
a·sinB = b·sinA
Divide both sides by sinB
sinB = b·sinA/a
sinB = (22.1)sin(33°46')/8.4
sinB = 1.4623151
Uh oh! The sine of an angle cannot exceed 1; therefore
there is no solution. That means side a, which is 8.4,
is too short to reach down to AB, and we cannot draw a
triangle with those parts.
Edwin
AnlytcPhil@aol.com
