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

Answer

To determine whether \( \tan \theta \) increases or decreases as \( \theta \) decreases 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 angle is \( \theta \), and the right angle is at vertex \( B \). Let the sides opposite and adjacent to angle \( \theta \) be labeled as follows:
- Opposite side (perpendicular) = \( a \)
- Adjacent side (base) = \( b \)
- Hypotenuse = \( c \)

2. **Definition of Tangent**:
- The tangent of angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side:
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{a}{b} \]

3. **Effect of Decreasing \( \theta \)**:
- As \( \theta \) decreases, the angle \( \theta \) becomes smaller. This means that the opposite side \( a \) (which corresponds to the height of the triangle) will also decrease while the adjacent side \( b \) remains constant (as it is the base of the triangle).
- Since \( a \) decreases, the ratio \( \frac{a}{b} \) will also decrease.

4. **Conclusion**:
- Therefore, we can conclude that as \( \theta \) decreases, \( \tan \theta \) also decreases. Hence, we can state:
\[ \text{If } \theta \text{ decreases, then } \tan \theta \text{ decreases.} \]

### Final Answer:
As \( \theta \) decreases, \( \tan \theta \) decreases.

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