Chapter 8.6, Problem 45PPS

a.

To determine

Verify the statement for the derivation of the law of sines.

Answer to Problem 45PPS

  $Definitionof\mathrm{sine}$

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{CD}isanaltitudeof\Delta ABC\\ \mathrm{Prove}:\frac{\sin A}{a}=\frac{\sin B}{b}\\ \mathrm{Proof}:\sin A=\frac{h}{b},\sin B=\frac{h}{a}\end{array}$

Calculation:

Here, we will consider the following figure:

  

Now, if $\Delta ABC$ is a right triangle with acute $\angle A$ is the ratio of the length of the leg opposite $\angle A$ to the length of hypotenuse.

Thus, $\sin A=\frac{h}{b}and\sin B=\frac{h}{a}$

Hence, the required answer is $Definitionof\mathrm{sine}$ .

b.

To determine

Verify the statement for the derivation of the law of sines.

Answer to Problem 45PPS

  $Multiplicationproperty$

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{CD}isanaltitudeof\Delta ABC\\ \mathrm{Prove}:\frac{\sin A}{a}=\frac{\sin B}{b}\\ \mathrm{Proof}:b\sin A=h,a\sin B=h\end{array}$

Calculation:

Here, we will consider the following figure:

  

  $\begin{array}{l}Multiplyeachsideof\sin A=\frac{h}{b}byb:\\ b\sin A=h\\ Multiplyeachsideof\mathrm{sinB}=\frac{h}{a}bya:\\ asinB=h\end{array}$

Hence, the required answer is $Multiplicationproperty$ .

c.

To determine

Verify the statement for the derivation of the law of sines.

Answer to Problem 45PPS

  $Substitutionproperty$

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{CD}isanaltitudeof\Delta ABC\\ \mathrm{Prove}:\frac{\sin A}{a}=\frac{\sin B}{b}\\ \mathrm{Proof}:b\sin A=a\sin B\end{array}$

Calculation:

Here, we will consider the following figure:

  

  $\begin{array}{l}Substitutinga\sin B=hinb\sin A=h:\\ b\sin A=a\sin B\end{array}$

Hence, the required answer is $Substitutionproperty$ .

d.

To determine

Verify the statement for the derivation of the law of sines.

Answer to Problem 45PPS

  $Multiplicationproperty$

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{CD}isanaltitudeof\Delta ABC\\ \mathrm{Prove}:\frac{\sin A}{a}=\frac{\sin B}{b}\\ \mathrm{Proof}:\frac{\sin A}{a}=\frac{\sin B}{b}\end{array}$

Calculation:

Here, we will consider the following figure:

  

  $\begin{array}{l}Multiplyeachsideofb\mathrm{sinA}=a\sin Bby\frac{1}{ab}:\\ \frac{\sin A}{a}=\frac{\sin B}{b}\end{array}$

Hence, the required answer is $Multiplicationproperty$ .