Question

law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$
in $\triangle abc$, $c = 5.4$, $a = 3.3$, and $mangle a=20^{circ}$. 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

Explanation:

Step1: Apply the law of sines

By the law of sines, $\frac{\sin(A)}{a}=\frac{\sin(C)}{c}$. We know $a = 3.3$, $c=5.4$, and $A = 20^{\circ}$. First, find $\sin(C)$: $\sin(C)=\frac{c\sin(A)}{a}=\frac{5.4\times\sin(20^{\circ})}{3.3}=\frac{5.4\times0.342}{3.3}\approx0.56$.

Step2: Find the two possible values of angle C

Since $\sin(C)\approx0.56$, $C_1=\sin^{- 1}(0.56)\approx34^{\circ}$ and $C_2 = 180^{\circ}-34^{\circ}=146^{\circ}$.

Step3: Find the two possible values of angle B

For $C_1 = 34^{\circ}$, $B_1=180^{\circ}-A - C_1=180^{\circ}-20^{\circ}-34^{\circ}=126^{\circ}$. For $C_2 = 146^{\circ}$, $B_2=180^{\circ}-A - C_2=180^{\circ}-20^{\circ}-146^{\circ}=14^{\circ}$.

Step4: Use the law of sines to find b for each case

Case 1: When $B = B_1 = 126^{\circ}$, $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}$, so $b_1=\frac{a\sin(B_1)}{\sin(A)}=\frac{3.3\times\sin(126^{\circ})}{\sin(20^{\circ})}=\frac{3.3\times0.809}{0.342}\approx7.8$.
Case 2: When $B = B_2 = 14^{\circ}$, $b_2=\frac{a\sin(B_2)}{\sin(A)}=\frac{3.3\times\sin(14^{\circ})}{\sin(20^{\circ})}=\frac{3.3\times0.242}{0.342}\approx2.3$.

Answer:

C. 2.3 units and 7.8 units