Given:
The sides of the triangle are given as,
[tex]\begin{gathered} a\text{ = 4 m} \\ b\text{ = 7 m} \\ c\text{ = 8 m} \end{gathered}[/tex]Required:
The smallest angle in a given triangle.
Explanation:
The smallest angle lie across the smallest side.
By using cosine rule,
[tex]a^2\text{ = b}^2+c^2-2bc.cos(\alpha)[/tex]Substituting the values in the given expression,
[tex]\begin{gathered} 4^2\text{ = 7}^2+8^2-2\times7\times8\times cos(\alpha) \\ 16\text{ = 49 + 64 - 112cos\lparen}\alpha) \\ 16\text{ = 113-112cos\lparen}\alpha) \\ \end{gathered}[/tex]Calculating the value of the smallest angle,
[tex]\begin{gathered} 112cos(\alpha)\text{ = 113-16} \\ 112cos(\alpha)\text{ = 97} \\ cos(\alpha)\text{ = }\frac{97}{112} \\ cos(\alpha)\text{ = 0.8661} \\ \alpha=\text{ cos}^{-1}(0.8661) \\ \alpha\text{ = 29.99}\approx\text{ 30} \end{gathered}[/tex]Answer:
Thus the measure of the smallest angle is 30 degrees.
