Respuesta :
Use the law of sines to find the measure of the angle B. Use that information to find the angle C using the fact that the internal angles of any triangle add up to 180°. Finally, use the measure of C and the law of sines again to find the length of the side c.
From the law of sines, we know that:
[tex]\begin{gathered} \frac{\sin(A)}{a}=\frac{\sin (B)}{b} \\ \Rightarrow\sin (B)=\frac{b}{a}\sin (A) \\ \Rightarrow B=\sin ^{-1}(\frac{b}{a}\sin (A)) \end{gathered}[/tex]Substitute b=28, a=32 and A=49 to find B. Use a calculator to find the value of B:
[tex]\begin{gathered} \Rightarrow B=\sin ^{-1}(\frac{28}{32}\sin (49)) \\ =\sin ^{-1}(\frac{7}{8}\sin (49)) \\ =\sin ^{-1}(0.6603708827\ldots) \\ =41.32816455\ldots \end{gathered}[/tex]Since the internal angles of a triangle add up to 180°, then:
[tex]\begin{gathered} A+B+C=180 \\ \Rightarrow C=180-A-B \\ \Rightarrow C=180-49-41.328\ldots \\ \Rightarrow C=89.67183545\ldots \end{gathered}[/tex]Use the law of sines again to find c:
[tex]\begin{gathered} \frac{a}{\sin(A)}=\frac{c}{\sin (C)} \\ \Rightarrow c=\frac{\sin(C)}{\sin(A)}\times a \\ =\frac{\sin(89.6718\ldots)}{\sin(49)}\times32 \\ =\frac{0.99998\ldots}{0.7547\ldots}\times32 \\ =1.32499\ldots\times32 \\ =42.3997\ldots \end{gathered}[/tex]To the nearest tenth:
[tex]c=42.4[/tex]