Chapter 8.2, Problem 6AYU

True or False An oblique triangle in which two sides and an angle are given always results in at least one triangle.

To determine

To find: Whether True or False: An oblique triangle in which two sides and an angle are given always results in at least one triangle.

Answer to Problem 6AYU

Solution:

False.

Explanation of Solution

Given:

An oblique triangle in which two sides and an angle are given always results in at least one triangle.

Calculation:

Let $a, b$, and $B$ be known, and let $B$ be obtuse. Using the Law of Sines, $\text{sin}A = \frac{\alpha \text{sin}B}{b}$. different cases exist.

1. If the side opposite the given angle is less than the other given side $(b < a)$, then ${\text{sin}}^{ -1}\left(\frac{\alpha \text{sin}(B)}{b}\right) + B > {180}^{\circ }$, so there is no solution, and no triangle is determined.

2. If the side opposite the given angle is equal to the other given side $(b = a)$, then ${\text{sin}}^{ -1}\left(\frac{\alpha \text{sin}(B)}{b}\right) + B = {180}^{\circ }$, so there is no solution, and, again, no triangle is determined.

3. If the side opposite the given angle is greater than the other given side, then exactly one triangle is determined.

Hence the statement “An oblique triangle in which two sides and an angle are given always results in at least one triangle” is false.