Chapter 8.2, Problem 32AYU

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).

$a=3$, $b=7$, $A={70}^{\circ }$

To determine

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

Answer to Problem 32AYU

Solution:

The given information results in no triangle.

Explanation of Solution

Given:

$\text{a} = 3, \text{b} = 7$ and $\text{A} = {70}^{\circ }$

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:

Two sides and one angle are known, use the law of sines to find the angle $\text{B}$

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

$\frac{\text{sin}70°}{3} = \frac{\text{sin}\text{B}}{7}$

$\text{sin}\text{B} = \frac{7\text{sin}70°}{3} \approx 2.19$

$\text{sin}$ value is never greater than 1 so from the given values we can’t form a triangle.