Question

in △abc, c = 5.4, a = 3.3, and m∠a = 20°. what are the possible approximate lengths of b? use the law of sines to find the answer.
2.0 units and 4.6 units
2.1 units and 8.7 units
2.3 units and 7.8 units
2.6 units and 6.6 units
law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Use the law of sines to find $\sin C$

By the law of sines $\frac{\sin A}{a}=\frac{\sin C}{c}$. Given $a = 3.3$, $c=5.4$ and $A = 20^{\circ}$, we have $\sin C=\frac{c\sin A}{a}$. Substituting the values: $\sin C=\frac{5.4\times\sin(20^{\circ})}{3.3}=\frac{5.4\times0.342}{3.3}\approx0.562$.

Step2: Find the two possible values of $C$

Since $\sin C\approx0.562$, $C_1=\sin^{- 1}(0.562)\approx34.2^{\circ}$ and $C_2 = 180^{\circ}-34.2^{\circ}=145.8^{\circ}$.

Step3: Find the two possible values of $B$

For $C_1 = 34.2^{\circ}$, $B_1=180^{\circ}-A - C_1=180^{\circ}-20^{\circ}-34.2^{\circ}=125.8^{\circ}$. For $C_2 = 145.8^{\circ}$, $B_2=180^{\circ}-A - C_2=180^{\circ}-20^{\circ}-145.8^{\circ}=14.2^{\circ}$.

Step4: Use the law of sines to find the two possible values of $b$

Using $\frac{\sin A}{a}=\frac{\sin B}{b}$, we get $b=\frac{a\sin B}{\sin A}$.
For $B_1 = 125.8^{\circ}$, $b_1=\frac{3.3\times\sin(125.8^{\circ})}{\sin(20^{\circ})}=\frac{3.3\times0.814}{0.342}\approx7.8$.
For $B_2 = 14.2^{\circ}$, $b_2=\frac{3.3\times\sin(14.2^{\circ})}{\sin(20^{\circ})}=\frac{3.3\times0.245}{0.342}\approx2.3$.

Answer:

C. 2.3 units and 7.8 units