Question

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

Explanation:

Step1: Identify sides for angle M

In right triangle \( MNL \) with right angle at \( N \), for angle \( M \):

• Opposite side (\( \text{opp} \)) to \( M \) is \( NL = 15 \)
• Hypotenuse (\( \text{hyp} \)) is \( ML = 25 \)
• Adjacent side (\( \text{adj} \)) to \( M \) can be found using Pythagoras: \( MN=\sqrt{ML^{2}-NL^{2}}=\sqrt{25^{2}-15^{2}}=\sqrt{625 - 225}=\sqrt{400}=20 \)

Step2: Recall cosine definition

Cosine of an angle in a right triangle is \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \)

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

For angle \( M \), adjacent side is \( MN = 20 \), hypotenuse is \( ML = 25 \)
So, \( \cos(M)=\frac{MN}{ML}=\frac{20}{25}=\frac{4}{5} \)

Answer:

\(\frac{4}{5}\) (assuming the option with \(\frac{4}{5}\) is one of the choices, e.g., if the options are as per the image, the correct option with \(\frac{4}{5}\))