Question

find the length of side c.

b
40°
c
a 120° 8 c

c = 20° a = 10.8 c = ?

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that in any triangle, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

Step2: Identify known values

We know that $a = 10.8$, $A = 120^\circ$, $C = 20^\circ$, and we need to find $c$.

Step3: Apply the Law of Sines

Using the Law of Sines, we have $\frac{c}{\sin C}=\frac{a}{\sin A}$.
Substitute the known values: $\frac{c}{\sin 20^\circ}=\frac{10.8}{\sin 120^\circ}$.

Step4: Solve for $c$

First, calculate $\sin 20^\circ\approx0.3420$ and $\sin 120^\circ=\sin(60^\circ)=\frac{\sqrt{3}}{2}\approx0.8660$.
Then, $c=\frac{10.8\times\sin 20^\circ}{\sin 120^\circ}$.
Substitute the values: $c=\frac{10.8\times0.3420}{0.8660}$.
Calculate the numerator: $10.8\times0.3420 = 3.6936$.
Then, $c=\frac{3.6936}{0.8660}\approx4.265$.
Rounding to a reasonable decimal place (assuming to the nearest tenth), $c\approx4.3$.

Answer:

$\approx 4.3$ (or more precise value depending on rounding requirements, e.g., if rounded to two decimal places, $\approx 4.27$)