Question

Solve the triangle

Short answer

The solution is

$a=127.17°,B=32.86°,C=19.97°$

Step-by-step solution

Step.1 Given information

Step2. Formula used

$$\begin{gathered} \cos A=\frac{b^2+c^2-a^2}{2bc}, \\ \cos B=\frac{a^2+c^2-b^2}{2ac} \end{gathered}$$

Step3. Calculation

$$\begin{gathered} Weknow, \\ \cos A=\frac{b^2+c^2-a^2}{2bc} \\ =\frac{6^2+4^2-9^2}{2.6.4} \\ =\frac{36+16-81}{48} \\ =\frac{-29}{48} \\ =-.6042 \\ A={\cos }^{-1}\left(-.6042\right) \\ =127.17° \\ Again \\ \cos B=\frac{a^2+c^2-b^2}{2ac} \\ =\frac{9^2+4^2-6^2}{2.9.4} \\ =\frac{81+16-36}{72} \\ =\frac{61}{72} \\ =.84 \\ B={\cos }^{-1}\left(.84\right) \\ =32.86° \\ Againweknow, \\ A+B=C=180° \\ 127.17°+32.86°+C=180° \\ C=180°-160.03 \\ =19.97° \end{gathered}$$