Part a) In ΔABC, c = 5.4, a = 3.3, and m∠A=20° . What are the possible approximate lengths of b? Use the law of sines to find the answer.
we know that
\(\frac{sin\ A}{a} =\frac{sin\ B}{b} =\frac{sin\ C}{c}\)
Step \(1\)
Find the value of angle C
\(\frac{sin\ A}{a} =\frac{sin\ C}{c}\)
\(\frac{sin\ 20}{3.3} =\frac{sin\ C}{5.4}\\ \\ sin\ C=\frac{5.4*sin\ 20}{3.3} \\ \\ sin\ C=0.5597\\ \\ C=arcsin(0.5597)\\ \\ C=34\ degrees\)
Step \(2\)
Find the value of angle B
we know that
\(A+B+C=180\\ B=180-(A+C)\\ B=180-(20+34)\\ B=126\ degrees\)
Step \(3\)
Find the value of side b
\(\frac{sin\ A}{a} =\frac{sin\ B}{b}\)
\(\frac{sin\ 20}{3.3} =\frac{sin\ 126}{b}\\ \\ b=\frac{3.3*sin\ 126}{sin\ 20} \\ \\ b=7.8\ units\)
Step \(4\)
Find the alternative angle C
\(C=180-34\\ C=146\ degrees\)
Find the alternative angle B
\(A+B+C=180\\ B=180-(A+C)\\ B=180-(20+146)\\ B=14\ degrees\)
Find the alternative value of side b
\(\frac{sin\ A}{a} =\frac{sin\ B}{b}\)
\(\frac{sin\ 20}{3.3} =\frac{sin\ 14}{b}\\ \\ b=\frac{3.3*sin\ 14}{sin\ 20} \\ \\ b=2.3\ units\)
therefore
the answer Part a) is the option
\(C:\ 2.3\ units\ and\ 7.8\ units\)
Part b) What is the approximate value of k? Use the law of sines to find the answer
Step \(1\)
Find the value of angle J
we know that
\(J+K+L=180\\ J=180-(K+L)\\ J=180-(120+40)\\ J=20\ degrees\)
Step \(2\)
Find the value of side k
\(\frac{sin\ K}{k} =\frac{sin\ J}{j}\)
\(\frac{sin\ 120}{k} =\frac{sin\ 20}{2}\\ \\ k=\frac{2*sin\ 120}{sin\ 20} \\ \\ k=5.1\ units\)
therefore
the answer Part b)
\(k=5.1\ units\)
