Question

law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$
in $\triangle fgh$, $h = 10$, $mangle f=65^{circ}$, and $mangle g = 35^{circ}$. what is the length of $g$? use the law of sines to find the answer.
5.8 units
6.7 units
9.2 units
9.8 units

Explanation:

Step1: Find angle $H$

The sum of angles in a triangle is $180^{\circ}$. So $m\angle H=180^{\circ}-(m\angle F + m\angle G)=180^{\circ}-(65^{\circ}+ 35^{\circ}) = 80^{\circ}$.

Step2: Apply the law of sines

According to the law of sines $\frac{\sin(G)}{g}=\frac{\sin(H)}{h}$. We know $h = 10$, $m\angle G=35^{\circ}$, $m\angle H = 80^{\circ}$. Rearranging for $g$ gives $g=\frac{h\sin(G)}{\sin(H)}$.

Step3: Calculate $g$

$g=\frac{10\times\sin(35^{\circ})}{\sin(80^{\circ})}$. Since $\sin(35^{\circ})\approx0.574$ and $\sin(80^{\circ})\approx0.985$, then $g=\frac{10\times0.574}{0.985}\approx5.8$.

Answer:

5.8 units