Question

two sides and an angle are given below. determine whether the given information results in one triangle, two triangles, or no triangle at all. solve any resulting triangle(s). b = 8, c= 7, b = 170°

Answer 1

A. single triangle. C = 8.74°, A = 1.26°, a = 1.01

Explanation:

When two sides and an angle are given, the possibility of two (or zero) triangles exists only when the given angle is opposite the shorter of the two given sides. That is not the case here, so the given measures define one unique triangle.

Law of Sines

The law of sines tells us the sides and angles have the relation ...

sin(A)/a = sin(B)/b = sin(C)/c

We can use this first to find angle C:

sin(C) = c/b·sin(B)

C = arcsin(c/b·sin(B)) = arcsin(7/8·sin(170°)) ≈ 8.7394°

∠C ≈ 8.74°

Remaining measures

Now that we know two angles, we can find the third:

A = 180° -B -C = 180° -170° -8.74° = 1.26°

∠A = 1.26°

Using the law of sines again, we can find the measure of side 'a'.

a = b·sin(A)/sin(B) = 8·sin(1.26°)/sin(170°) ≈ 1.01346

The measure of segment 'a' is about 1.01 units.

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Additional comment

If 'a', 'b', and angle A are given, there will be zero triangles if b/a·sin(A) > 1. If b/a·sin(A) < 1 and b > a, there will be two (2) triangles. Otherwise, as here, there will be one unique triangle.