Chapter 6.5, Problem 21E

To determine

To calculate: The complete triangle with the help of law of sines if possible with the given conditions.

Answer to Problem 21E

There is no solution possible of triangle that satisfies given conditions.

Explanation of Solution

Given information:

  $a=20,c=45\text{ and }\angle {\text{A=125}}^{\circ }$ .

Formula used:

According to law of sines, the lengths of the sides of the triangle are proportional to the sines of the corresponding opposite angles. In any triangle, $\Delta \text{XYZ}$

  $\frac{\text{sin X}}{\text{x}}=\frac{\text{sin Y}}{\text{y}}\text{=}\frac{\text{sin Z}}{\text{z}}$

Calculation:

Consider the given values, $a=20,c=45\text{ and }\angle {\text{A=125}}^{\circ }$ .

Recall that according to law of sines, the lengths of the sides of the triangle are proportional to the sines of the corresponding opposite angles. In any triangle, $\Delta \text{XYZ}$

  $\frac{\text{sin X}}{\text{x}}=\frac{\text{sin Y}}{\text{y}}\text{=}\frac{\text{sin Z}}{\text{z}}$

So, the value of $\angle \text{C}$ will be calculated as,

  $\begin{array}{c}\frac{\text{sin A}}{a}=\frac{\text{sin C}}{c}\\ \frac{\sin 125}{20}=\frac{\sin \text{C}}{45}\\ \sin C=\frac{45\sin 125}{20}\\ \text{sinC}\approx 1.84\end{array}$

Since, sine of any angle cannot exceed 1 i.e. $-1\le \sin \theta \le 1$ ,

So, $\sin C$ is more than 1, which is not possible,

Thus, there exists no triangle that satisfies the given conditions, so, there is no solution possible of triangle that satisfies given conditions.