Question

what is the approximate value of b, rounded to the nearest tenth? use the law of sines to find the answer. law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$ 4.7 units 5.0 units

Explanation:

Step1: Find angle C

The sum of angles in a triangle is 180°. So $C=180^{\circ}-66^{\circ}-75^{\circ}=39^{\circ}$.

Step2: Apply law of sines

We know that $\frac{\sin A}{a}=\frac{\sin B}{b}$. Here $A = 66^{\circ}$, $a = 3$, $B=75^{\circ}$, and we want to find $b$. So $b=\frac{a\sin B}{\sin A}$.

Step3: Substitute values

$b=\frac{3\times\sin(75^{\circ})}{\sin(66^{\circ})}$. Since $\sin(75^{\circ})=\sin(45^{\circ} + 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.966$ and $\sin(66^{\circ})\approx0.914$. Then $b=\frac{3\times0.966}{0.914}\approx3.15\div0.914\approx 4.7$.

Answer:

4.7 units