Question

given right triangle mnl, what is the value of cos(m)? options: 3/5, 3/4, 4/5, 5/3

Explanation:

Step1: Recall the cosine formula

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\angle L\), the adjacent side to \(\angle L\) is \(NL\) and the hypotenuse is \(ML\).

Step2: Use the Pythagorean theorem to find \(NL\)

By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 25\) (hypotenuse \(ML\)) and \(b = 15\) (\(MN\)). Let \(a=NL\). Then \(NL=\sqrt{ML^{2}-MN^{2}}=\sqrt{25^{2}-15^{2}}=\sqrt{(25 + 15)(25 - 15)}=\sqrt{40\times10}=\sqrt{400}=20\).

Step3: Calculate \(\cos\angle L\)

\(\cos\angle L=\frac{NL}{ML}\). Since \(NL = 20\) and \(ML=25\), \(\cos\angle L=\frac{20}{25}=\frac{4}{5}\).

Answer:

\(\frac{4}{5}\)