Question

what is the approximate measure of angle k? use the law of sines to find the answer. law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$ 20° 34° 41° 53°

Explanation:

Step1: Apply law of sines

According to the law of sines $\frac{\sin(L)}{j}=\frac{\sin(K)}{l}$, where $L = 105^{\circ}$, $j = 2.7$, and $l=4.7$. So $\frac{\sin(105^{\circ})}{2.7}=\frac{\sin(K)}{4.7}$.

Step2: Solve for $\sin(K)$

Cross - multiply to get $\sin(K)=\frac{4.7\times\sin(105^{\circ})}{2.7}$. First, $\sin(105^{\circ})=\sin(60^{\circ}+ 45^{\circ})=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659$. Then $\sin(K)=\frac{4.7\times0.9659}{2.7}\approx1.689\times0.9659\approx0.656$.

Step3: Find angle $K$

$K=\sin^{- 1}(0.656)$. Using a calculator, $K\approx41^{\circ}$.

Answer:

$41^{\circ}$