Question

j k 4.7 2.7 105° l law of sines: (sin(a))/a = (sin(b))/b = (sin(c))/c what is the approximate measure of angle k? use the law of sines to find the answer. 20° 34° 41° 53°

Explanation:

Step1: Apply the Law of Sines

In \(\triangle JKL\), by the Law of Sines \(\frac{\sin L}{JK}=\frac{\sin K}{JL}\). Given \(L = 105^{\circ}\), \(JK = 4.7\), and \(JL=2.7\). So \(\frac{\sin(105^{\circ})}{4.7}=\frac{\sin K}{2.7}\).

Step2: Solve for \(\sin K\)

Cross - multiply: \(\sin K=\frac{2.7\times\sin(105^{\circ})}{4.7}\). Since \(\sin(105^{\circ})=\sin(60^{\circ} + 45^{\circ})=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659\). Then \(\sin K=\frac{2.7\times0.9659}{4.7}\approx\frac{2.608}{4.7}\approx0.555\).

Step3: Find the angle \(K\)

Take the inverse - sine: \(K=\sin^{- 1}(0.555)\approx34^{\circ}\).

Answer:

\(34^{\circ}\) (the second option)