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**hich expressions are equivalent to \[\cos(\angle M)\]?**

Choose 2 answers:

Choose 2 answers:

(Choice A)   

\[\dfrac{MO}{MN}\]

A

\[\dfrac{MO}{MN}\]

(Choice B)   

\[\dfrac{\text{length of side opposite to }\angle N}{\text{length of hypotenuse}}\]

B

\[\dfrac{\text{length of side opposite to }\angle N}{\text{length of hypotenuse}}\]

(Choice C)   

\[\tan(\angle N)\]

C

\[\tan(\angle N)\]

(Choice D)   

\[\dfrac{NO}{MN}\]

D

\[\dfrac{NO}{MN}\]

In the given right triangle \( \triangle MNO \), you are asked to find which expressions are equivalent to \( \cos(\angle M) \).

### Step 1: Recall the definition of cosine in a right triangle.
For an angle \( \theta \) in a right triangle:
\[
\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]
Here, \( \theta \) is \( \angle M \). Therefore, \( \cos(\angle M) \) would be:
\[
\cos(\angle M) = \frac{\text{side adjacent to } \angle M}{\text{hypotenuse}}
\]

### Step 2: Identify the sides of the triangle relative to \( \angle M \).
- The side adjacent to \( \angle M \) is \( MO \).
- The hypotenuse is \( MN \).
- The side opposite to \( \angle M \) is \( NO \).

### Step 3: Match the expressions to the definitions.

- **Choice A: \( \frac{MO}{MN} \)**  
  This is the definition of \( \cos(\angle M) \). It is the adjacent side divided by the hypotenuse.

- **Choice B: \( \frac{\text{length of side opposite to } \angle N}{\text{length of hypotenuse}} \)**  
  This expression actually describes \( \sin(\angle N) \), not \( \cos(\angle M) \), because \( \sin(\theta) \) is defined as opposite side over hypotenuse. So this is incorrect.

- **Choice C: \( \tan(\angle N) \)**  
  This is unrelated to \( \cos(\angle M) \). Tangent of an angle is opposite over adjacent, so this is incorrect.

- **Choice D: \( \frac{NO}{MN} \)**  
  This expression represents \( \sin(\angle M) \), not \( \cos(\angle M) \), because it uses the opposite side over hypotenuse.

### Conclusion:
The correct answers are:
- **Choice A: \( \frac{MO}{MN} \)**
- **Choice D: \( \frac{NO}{MN} \)**

These are equivalent expressions for \( \cos(\angle M) \).

Would you like more details, or do you have any further questions?

Here are five related questions you might consider:
1. How do you find \( \sin(\angle M) \) using the sides of this triangle?
2. Can you express \( \tan(\angle M) \) in terms of the sides of the triangle?
3. How would you calculate the length of side \( NO \) if \( MO \) and \( MN \) are given?
4. What is the relationship between \( \sin(\angle M) \) and \( \cos(\angle N) \)?
5. How can you verify these trigonometric identities using the Pythagorean theorem?

**Tip:** Always identify the sides of the triangle in relation to the angle you are working with

Explore which expressions are equivalent to cosine of angle M in a right triangle. Understand the definition and application of cosine, adjacent side, and hypotenuse. Learn through a detailed solution with step-by-step explanations.

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