Question 1126585: Find all solutions to the following triangle. (Round your answers to one decimal place. If either triangle is not possible, enter NONE in each corresponding answer blank.)
A = 119.1°, a = 43.9 cm, b = 25.1 cm
First triangle (assume B ≤ 90°):
B= °
C= °
c= cm
Second triangle (assume B' > 90°):
B'= °
C'= °
c'= cm
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
In this kind of problem, you are always given one angle and the length of the side opposite that angle, plus the length of one of the two other sides.
I find it easiest always to draw the figure in the same orientation: unknown side horizontal, with the given angle at the left.
So in this problem the horizontal base of the triangle is AB, with the 119.1 degree angle A at the left. (We don't know yet where B will be.) Then the 25.1cm side b slants up to the left from A to C; side a (i.e., CB) of length 43.9cm slants down to the right from C.
Since the 43.9 is longer than the 25.1, it should be clear from a rough sketch that there will be one triangle for this given set of measurements.
To solve the triangle, you can do the following:
(1) Find the altitude of the triangle (the vertical distance from C to side AB extended). That length (call it d) is b*sin(A) = 25.1*sin(119.1).
(2) Angle B is then the inverse sine of (d/a).
(3) Then find angle C knowing that the sum of the three angles is 180 degrees.
(4) Finally, find the length of AB (side c) using either the law of sines or the law of cosines.
As a check of the calculations you make, I will tell you that the altitude d turns out to be almost exactly half of a; that makes angle B very close to 30 degrees (sin(30) = 1/2). That, with angle A being 119.1 degrees, makes triangle ABC very nearly isosceles; so the length you find for c should be very close to the given 25.1cm length of side b.
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