Question

why can't you solve for side b as your first step to this problem?

Answer 1

The triangle in the question can be solved using two rules

They are:

SINE RULE AND COSINE RULE

The sine rule states that

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

While the cosine rule is

$\begin{gathered} a^2=b^2+c^2-2bc\cos A \\ b^2=a^2+c^2-2\text{ac}\cos B \\ c^2=a^2+b^2-2ab\cos C \end{gathered}$

The given values in the question include

$\begin{gathered} a=8.4,A=26^0 \\ b=\text{?,B}=\text{?} \\ c=12.4,C=? \end{gathered}$

We cannot solve b as the first step in the question because the angle at B is not given

Step Instead, we will have to, first of all, use the sine rule below to get the angle at C

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

Substituting the values, we will have

$\begin{gathered} \frac{a}{\sin A}=\frac{c}{\sin C} \\ \frac{8.4}{\sin26^0}=\frac{12.4}{\sin C} \end{gathered}$

Cross multiply, we will have

$\begin{gathered} \frac{8.4}{\sin26^0}=\frac{12.4}{\sin C} \\ 8.4\times\sin C=12.4\times\sin 26^0 \\ 8.4\sin C=5.4358 \\ \text{divide both sides by 8.4} \\ \frac{8.4\sin C}{8.4}=\frac{5.4358}{8.4} \\ \sin C=0.6471 \\ C=\sin ^{-1}0.6471 \\ C=40.3^0 \end{gathered}$

Step 2: Calculate the value of angle B next using the sum of angles in a triangle=180°

$\begin{gathered} \angle A+\angle B+\angle C=180^0 \\ 26^0+\angle B+40.3^0=180^0 \\ 66.3^0+\angle B=180^0 \\ \angle B=180^0-66.3 \\ \angle B=113.7^0 \end{gathered}$

Step 3: Calculate the value of b using the the sine rule below

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

Substituting the value, we will have

$\begin{gathered} \frac{a}{\sin A}=\frac{b}{\sin B} \\ \frac{8.4}{\sin26^0}=\frac{b}{\sin113.7^0} \end{gathered}$

Cross multiply, we will have

$\begin{gathered} \frac{8.4}{\sin26^0}=\frac{b}{\sin113.7^0} \\ b\times\sin 26=8.4\times\sin 113.7^0 \\ b=\frac{8.4\times\sin 113.7}{\sin 26^0} \\ b=17.5 \end{gathered}$

Therefore,

We cannot solve b as the first step because no side (b) or angle for B was given

We will have to calculate the value of Angle C(because the side at c was given) , followed by calculating the value of angle B using the sum of angles in a triangle (180) and then calculate the value of b using the sine rule (because the angle at B has been calculated)

The explanation of the solution is shown above