Question

19
the sides of a triangle are 6, 10, and 14. compute the value of the cosine of the
smallest angle of the triangle.

\\( \frac { 6 } { 7 } \\)
\\( \frac { 5 } { 7 } \\)
\\( \frac { 13 } { 14 } \\)
\\( \frac { 11 } { 14 } \\)

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In a triangle, the smallest angle is opposite the shortest side. Let \(a = 10\), \(b=14\), \(c = 6\). Then \(c^{2}=a^{2}+b^{2}-2ab\cos C\) (where \(C\) is the angle opposite side \(c\)).

Step2: Substitute the values into the formula

Substitute \(a = 10\), \(b = 14\), \(c = 6\) into \(c^{2}=a^{2}+b^{2}-2ab\cos C\). We get \(6^{2}=10^{2}+14^{2}-2\times10\times14\times\cos C\).

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Answer:

\(\frac{13}{14}\) (corresponding to the third option)