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# Solve the resulting triangle

Two sidea and an angle are given. Determine whether the given information results in no triangle, one triangle, or two triangle.

a=4 b=3 B=15degrees

### 1 Answer by Expert Tutors

John M. | Analytical assistance -- Writing, Math, and moreAnalytical assistance -- Writing, Math, ...
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We need to use the properties of the law of sines

You've been given a, b and B, but the page uses a, b, and A, so I'm going to swap things around and use
a=3, b=4 and B=15 degrees, so you can use the images on the page to follow along

If you scroll down the webpage to "The Ambiguous Case", you will see that there are 3 possible situations:  That no triangle can be created, that two different triangles can exist, or that exactly one triangle exists.  Also, we need to define "h" or the height of the supposed triangle, which is b sin A or 4 sin 15 degrees.

(1) Since A<90 degrees, A is acute, and thus no triangle can exist if a < 4 sin 15 degrees

(2) Since A is acute, two triangles exist if h<a<b

(3) Otherwise, since A is acute, only one triangle exists if a≥b

Case 1 does not apply, because  4 sin 15 = 1.04, and a = 3

Case 2 applies because 1.04<3<4, so there are two triangles.

Applying the law of sines
sin A/a = sin B/b -> b sin A/a = sin B or
4 sin 15 / 3 = 0.345, and two angles between 0-180 degrees produce a sine of .345: 20.18 degrees and 159.81 degrees (180-20.18 degrees).

I hope this helps.  John