Question

use the law of sines to find the value of a. what is the best approximation of the value of a? law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Find angle C

The sum of angles in a triangle is 180°. So, $C=180^{\circ}-40^{\circ}-95^{\circ}=45^{\circ}$.

Step2: Apply law of sines

According to the law of sines $\frac{\sin(A)}{a}=\frac{\sin(C)}{c}$. We know $c = 4.7$ cm, $A = 40^{\circ}$, $C=45^{\circ}$. Rearranging for $a$ gives $a=\frac{c\times\sin(A)}{\sin(C)}$.

Step3: Substitute values

$a=\frac{4.7\times\sin(40^{\circ})}{\sin(45^{\circ})}$. Since $\sin(40^{\circ})\approx0.6428$ and $\sin(45^{\circ})\approx0.7071$, then $a=\frac{4.7\times0.6428}{0.7071}\approx4.7\times0.9091\approx4.27$. This seems wrong. Let's use $\frac{\sin(B)}{b}=\frac{\sin(A)}{a}$ instead. Here $b = 4.7$ cm, $B = 95^{\circ}$, $A = 40^{\circ}$. So $a=\frac{b\times\sin(A)}{\sin(B)}$.

Step4: Recalculate

$\sin(A)= \sin(40^{\circ})\approx0.6428$, $\sin(B)=\sin(95^{\circ})\approx0.9962$, $b = 4.7$ cm. Then $a=\frac{4.7\times0.6428}{0.9962}\approx4.7\times0.6453\approx3.0$ cm.

Answer:

3.0 cm