Chapter 6.1, Problem 34E

(a)

To determine

The value of b for which the triangle has one solution $A={105}^0,a=80$ .

Answer to Problem 34E

The value of b for which the triangle has one solution is $b\le 80\text{ or }b=\frac{80}{\sin {105}^0}$ .

Explanation of Solution

Given info:

  $A={105}^0,a=80$

Formula used:

Law of sines

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

Calculation:

Given that $A={105}^0,a=80$ , We have

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{80}{\sin {105}^0}=\frac{b}{\sin B}\end{array}$

One solution if $b\le 80\text{ or }b=\frac{80}{\sin {105}^0}$

Conclusion:

Thus, the value of b for which the triangle has one solution is $b\le 80\text{ or }b=\frac{80}{\sin {105}^0}$ .

(b)

To determine

The value of b for which the triangle has two solutions $A={105}^0,a=80$ .

Answer to Problem 34E

The value of b for which the triangle has two solutions is $80\text{<}b<\frac{80}{\sin {105}^0}$.

Explanation of Solution

Given info:

  $A={105}^0,a=80$

Formula used:

Law of sines

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

Calculation:

Given that $A={105}^0,a=80$ , We have

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{80}{\sin {105}^0}=\frac{b}{\sin B}\end{array}$

Two solutions if $80\text{<}b<\frac{80}{\sin {105}^0}$

Conclusion:

Thus, the value of b for which the triangle has two solutions is $80\text{<}b<\frac{80}{\sin {105}^0}$

(c)

To determine

The value of b for which the triangle has no solutions $A={105}^0,a=80$ .

Answer to Problem 34E

The value of b for which the triangle has no solution is $b>\frac{80}{\sin {105}^0}$

Explanation of Solution

Given info:

  $A={105}^0,a=80$

Formula used:

Law of sines

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

Calculation:

Given that $A={105}^0,a=80$ , We have

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{80}{\sin {105}^0}=\frac{b}{\sin B}\end{array}$

No solutions if $b>\frac{80}{\sin {105}^0}$

Conclusion:

Thus, the value of b for which the triangle has no solution is $b>\frac{80}{\sin {105}^0}$