Question

find. \\(\cos n\\) \\(\triangle lmn\\) is a right triangle with right angle at \\(m\\), \\(lm = 5\\), \\(ln = 13\\)

Explanation:

Step1: Identify the triangle type

It's a right - triangle with right angle at \(M\), \(LM = 5\), \(LN=13\). First, find \(MN\) using Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = LN = 13\), \(a = LM = 5\), so \(MN=\sqrt{LN^{2}-LM^{2}}=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12\).

Step2: Recall the definition of cosine

In a right - triangle, \(\cos\theta=\frac{\text{adjacent side to } \theta}{\text{hypotenuse}}\). For \(\angle N\), the adjacent side is \(MN = 12\) and the hypotenuse is \(LN=13\). So \(\cos N=\frac{MN}{LN}=\frac{12}{13}\).

Answer:

\(\frac{12}{13}\)