Question

Solve the triangle

Short answer

The solution is$A=29.54°,C=105.46°,b=2.95$

Step-by-step solution

Step1. Given information

The figure is

Step2. Formula used

$$\begin{gathered} b^2=a^2+c^2-2ac\cos \left(B\right) \\ and\cos \left(A\right)=\frac{b^2+c^2-a^2}{2bc}and\cos \left(C\right)=\frac{a^2+b^2-c^2}{2ab} \end{gathered}$$

Step3. Calculation

We have

$$\begin{gathered} a=2,c=4,B=45° \\ weknow, \\ {b}^2=a^2+c^2-2ac\cos \left(B\right) \\ =2^2+4^2-2.2.4.\cos \left(45°\right) \\ =4+16-16.\frac{1}{\sqrt{2}} \\ =20-\frac{16.\sqrt[]{2}}{\sqrt{2.\sqrt{2}}} \\ =20-\frac{16\sqrt{2}}{2} \\ =20-8\sqrt{2} \\ =20-11.31 \\ =8.69 \\ b=\sqrt{8.69} \\ =2.95 \\ Again,\cos \left(A\right)=\frac{b^2+c^2-a^2}{2bc} \\ =\frac{2.{95}^2+4^2-2^2}{2\times 2.95\times 4} \\ =\frac{8.69+16-4}{23.6} \\ =\frac{20.69}{23.6} \\ =0.87 \\ A={\cos }^{-1}\left(0.87\right) \\ =29.54° \\ SinceA+B+C=180° \\ 29.54°+45°+C=180° \\ C=180°-74.54° \\ =105.46° \end{gathered}$$