Question

Solve each triangle using either the Law of Sines or the Law of Cosines

$a=6,b=8,c=9$

Short answer

The required triangle is

$$\begin{gathered} A=40.88° \\ B=60.59° \\ C=78.53° \end{gathered}$$

Step-by-step solution

Given information

$a=6,b=8,c=9$

To find the angles A,B,C

Calculation

$$\begin{gathered} ForA: \\ \cos \left(A\right)=\frac{b^2+c^2-a^2}{2bc} \\ =\frac{8^2+9^2-6^2}{2\times 8\times 9} \\ =0.756 \\ A={\cos }^{-1}\left(0.756\right) \\ =40.88° \end{gathered}$$

$$\begin{gathered} ForB: \\ \cos \left(B\right)=\frac{a^2+c^2-b^2}{2ac} \\ =\frac{6^2+9^2-8^2}{2\times 6\times 9} \\ =0.491 \\ B={\cos }^{-1}\left(0.491\right) \\ =60.59° \end{gathered}$$

Since we know A and B

$$\begin{gathered} C=180-A-B \\ =180°-40.88°-60.59° \\ =78.53° \end{gathered}$$