Question

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

Explanation:

Step1: Find side MN using Pythagorean theorem

In right - triangle MNL, by the Pythagorean theorem \(MN=\sqrt{ML^{2}-NL^{2}}\). Given \(ML = 25\) and \(NL=15\), then \(MN=\sqrt{25^{2}-15^{2}}=\sqrt{(25 + 15)(25 - 15)}=\sqrt{40\times10}=\sqrt{400}=20\).

Step2: Recall cosine definition

The cosine of an angle in a right - triangle is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\cos(M)\), the adjacent side to angle \(M\) is \(MN\) and the hypotenuse is \(ML\).

Step3: Calculate \(\cos(M)\)

\(\cos(M)=\frac{MN}{ML}\). Since \(MN = 20\) and \(ML=25\), then \(\cos(M)=\frac{20}{25}=\frac{4}{5}\).

Answer:

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