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.
2.1 units and 8.7 units
2.3 units and 7.8 units
2.0 units and 4.0 units
2.5 units and 6.6 units

Explanation:

Step1: Apply Law of Sines to find angle C

The Law of Sines states $\frac{\sin A}{a}=\frac{\sin C}{c}$. We know $a = 3.3$, $c = 5.4$, and $m\angle A=20^{\circ}$. So we can solve for $\sin C$:
$\sin C=\frac{c\sin A}{a}=\frac{5.4\times\sin(20^{\circ})}{3.3}$
First, calculate $\sin(20^{\circ})\approx0.3420$, then:
$\sin C=\frac{5.4\times0.3420}{3.3}=\frac{1.8468}{3.3}\approx0.5596$

Step2: Find possible measures of angle C

Since $\sin C\approx0.5596$, the reference angle is $C_1=\arcsin(0.5596)\approx34^{\circ}$. The other possible angle in the second quadrant is $C_2 = 180^{\circ}- 34^{\circ}=146^{\circ}$. We need to check if these angles are valid (i.e., the sum of angles in a triangle is $180^{\circ}$).

• For $C_1\approx34^{\circ}$: Then $m\angle B=180^{\circ}-20^{\circ}-34^{\circ}=126^{\circ}$
• For $C_2\approx146^{\circ}$: Then $m\angle B=180^{\circ}-20^{\circ}-146^{\circ}=14^{\circ}$

Step3: Apply Law of Sines to find side b for both cases

The Law of Sines also states $\frac{\sin A}{a}=\frac{\sin B}{b}$, so $b=\frac{a\sin B}{\sin A}$

• Case 1: $m\angle B = 126^{\circ}$, $\sin(126^{\circ})\approx\sin(180 - 54)^{\circ}=\sin(54^{\circ})\approx0.8090$

$b=\frac{3.3\times0.8090}{\sin(20^{\circ})}=\frac{3.3\times0.8090}{0.3420}\approx\frac{2.6697}{0.3420}\approx7.8$

• Case 2: $m\angle B = 14^{\circ}$, $\sin(14^{\circ})\approx0.2419$

$b=\frac{3.3\times0.2419}{\sin(20^{\circ})}=\frac{3.3\times0.2419}{0.3420}\approx\frac{0.79827}{0.3420}\approx2.3$

Answer:

2.3 units and 7.8 units