State for any acute angle `theta` whether :
`costheta` increases or decreases as `theta` increases.

Answer

To determine whether \( \cos \theta \) increases or decreases as \( \theta \) increases for any acute angle \( \theta \), we can analyze the relationship using a right triangle.

### Step-by-Step Solution:

1. **Understanding the Right Triangle**:
- Consider a right triangle where one of the angles is \( \theta \). Let's label the vertices as follows:
- Vertex A (the right angle)
- Vertex B (the angle \( \theta \))
- Vertex C (the remaining vertex)

2. **Identifying Sides**:
- In this triangle:
- The side opposite to angle \( \theta \) is called the opposite side (BC).
- The side adjacent to angle \( \theta \) is called the base (AB).
- The longest side, opposite the right angle, is called the hypotenuse (AC).

3. **Definition of Cosine**:
- The cosine of angle \( \theta \) is defined as:
\[ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{AB}{AC} \]

4. **Analyzing the Effect of Increasing \( \theta \)**:
- As \( \theta \) increases (while remaining an acute angle), the opposite side (BC) becomes longer, and the base (AB) becomes shorter.
- Consequently, the hypotenuse (AC) remains the longest side, but its length does not increase as significantly as the opposite side.

5. **Conclusion**:
- Since \( \cos \theta \) is the ratio of the base (AB) to the hypotenuse (AC), and as \( \theta \) increases, the base decreases while the hypotenuse remains relatively constant, the value of \( \cos \theta \) decreases.
- Therefore, we conclude that:
\[ \text{As } \theta \text{ increases, } \cos \theta \text{ decreases.} \]

### Final Answer:
As \( \theta \) increases, \( \cos \theta \) decreases for any acute angle \( \theta \).

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