Chapter 8.2, Problem 29AYU

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).

$a=4$, $b=5$, $A={60}^{\circ }$

To determine

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

Answer to Problem 29AYU

Solution:

No triangle will form.

Explanation of Solution

Given:

$\text{a} = 4, \text{b} = 5$ and $\text{A} = {60}^{\circ }$

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.

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

$\frac{\text{sin}60°}{4} = \frac{\text{sin}\text{B}}{5} = \frac{\text{sin}\text{C}}{\text{c}}$

$\frac{\text{sin}60°}{4} = \frac{\text{sin}\text{B}}{5}$

$\frac{\frac{\sqrt{3}}{2}}{4} = \frac{\text{sin}\text{B}}{5}$

$\text{sin}\text{B} = 5\left(\frac{\frac{\sqrt{3}}{2}}{4}\right)$

$\text{B} = {\text{sin}}^{ -1}(1.08)$

From above sin values are always less than or equal to 1 but not more than 1 so the given values are not of a triangle