Question

If A = 49 degrees, a = 32,and b = 28, find c.

Answer 1

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:

`\begin{gathered} (\sin(A))/(a)=(\sin (B))/(b) \\ \Rightarrow\sin (B)=(b)/(a)\sin (A) \\ \Rightarrow B=\sin ^(-1)((b)/(a)\sin (A)) \end{gathered}`

Substitute b=28, a=32 and A=49 to find B. Use a calculator to find the value of B:

`\begin{gathered} \Rightarrow B=\sin ^(-1)((28)/(32)\sin (49)) \\ =\sin ^(-1)((7)/(8)\sin (49)) \\ =\sin ^(-1)(0.6603708827\ldots) \\ =41.32816455\ldots \end{gathered}`

Since the internal angles of a triangle add up to 180°, then:

`\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}`

Use the law of sines again to find c:

`\begin{gathered} (a)/(\sin(A))=(c)/(\sin (C)) \\ \Rightarrow c=(\sin(C))/(\sin(A))* a \\ =(\sin(89.6718\ldots))/(\sin(49))*32 \\ =(0.99998\ldots)/(0.7547\ldots)*32 \\ =1.32499\ldots*32 \\ =42.3997\ldots \end{gathered}`

To the nearest tenth:

`c=42.4`