Question

Solve each triangle.

$a=10,b=8,c=5$

Short answer

The required answer is

$$\begin{gathered} A=81.95° \\ B=52.41° \\ C=45.64° \end{gathered}$$

Step-by-step solution

Given information

$a=10,b=8,c=5$

To find the angles A, B, C

Calculation

$$\begin{gathered} ForA: \\ \cos \left(A\right)=\frac{b^2+c^2-a^2}{2bc} \\ =\frac{8^2+5^2-{10}^2}{2\times 8\times 5} \\ =-0.14 \\ A={\cos }^{-1}\left(-0.14\right) \\ =81.95° \end{gathered}$$

$$\begin{gathered} ForB: \\ \cos \left(B\right)=\frac{a^2+c^2-b^2}{2ac} \\ =\frac{{10}^2+5^2-8^2}{2\times 10\times 5} \\ =0.61 \\ B={\cos }^{-1}\left(0.61\right) \\ =52.41° \end{gathered}$$

Since we know A and B

$$\begin{gathered} C=180°-A-B \\ =180°-81.95°-52.41° \\ =45.64° \end{gathered}$$