Chapter 8.2, Problem 28AYU

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).

$a=2$, $c=1$, $A={120}^{\circ }$

To determine

To find: The given information results in one triangle, two triangles, or no triangle at all.

Answer to Problem 28AYU

Solution:

One triangle will be formed

Explanation of Solution

Given:

$\text{a} = 2, \text{c} = 1, \text{A} = {120}^{\circ }$

Formula used:

$\frac{\text{sin}\text{A}}{\text{a}} = \frac{\text{sin}\text{B}}{\text{b}} = \frac{\text{sin}\text{C}}{\text{c}}$

$\angle \text{A} + \angle \text{B} + \angle \text{C} = {180}^{\circ }$

Calculation:

From sine law we know that

$\frac{\text{sin}\text{A}}{\text{a}} = \frac{\text{sin}\text{B}}{\text{b}} = \frac{\text{sin}\text{C}}{\text{c}}$

$\frac{\text{sin}50°}{2} = \frac{\text{sin}\text{B}}{\text{b}} = \frac{\text{sin}\text{C}}{1}$

$\frac{\text{sin}120°}{2} = \frac{\text{sin}\text{C}}{1}$

$\frac{\text{sin}120°}{2} = \text{ sinC}$

$\begin{array}{l}\text{SinC} = 0.433\\ \text{C} = {\text{sin}}^{ -1}(0.433)\\ \text{C} = {25.7}^{\circ }\end{array}$

There are two angles $\text{C}, 0^{\circ } < \text{C} < {180}^{\circ }$, for which $\text{A} = {120}^{\circ }$

${\text{C}}_1 = {25.7}^{\circ }$ and ${\text{C}}_2 = {180}^{\circ } - {25.7}^{\circ } = {154.3}^{\circ }$

The second possibility, ${\text{C}}_2 = {154.3}^{\circ }$ is ruled out because $\text{A} = {120}^{\circ }$ makes $\text{A} + {\text{B}}_2 = {180}^{\circ }$, now use ${\text{B}}_1 = {25.7}^{\circ }$ to find that

$\text{C} \approx {180}^{\circ } - \text{A} - {\text{B}}_1 = {180}^{\circ } - {120}^{\circ } - {25.7}^{\circ } \approx {34.3}^{\circ }$

The third side $\text{c}$ may now be determined using the Law of Sines.

$\frac{\text{sin}\text{A}}{\text{a}} = \frac{\text{sin}\text{C}}{\text{c}}$

$\frac{\text{sin}120°}{2} = \frac{\text{sin}34.3°}{\text{b}}$

$\text{b} = 2 \text{*} \frac{\text{sin}34.3°}{\text{sin}120°} \approx 1.3$

One triangle: $\text{B} \approx {34.3}^{\circ }, \text{C} = {25.7}^{\circ }, \text{b} \approx 1.3$