If the value , to the nearest thousandth , of tan `theta` is -2.747, which of the following could be true about `theta` ?

Options

A. `0 lt theta lt pi/4`
B. `pi/4 lt theta lt pi/2`
C. `pi/2 lt theta lt (3pi)/4`
D. `(5pi)/4 lt theta lt (3pi)/2`

Answer

To determine the possible values of \( \theta \) for which \( \tan \theta \) is approximately -2.747, we can analyze the behavior of the tangent function across its periodic intervals.

### Step-by-Step Solution:

1. **Understanding the Tangent Function**:
The tangent function, \( \tan \theta \), is periodic with a period of \( \pi \). It has vertical asymptotes at \( \theta = \frac{\pi}{2} + n\pi \) for any integer \( n \), where the function approaches infinity.

2. **Identifying the Range of Tangent**:
The tangent function takes on all real values between its vertical asymptotes. Specifically:
- From \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \), \( \tan \theta \) increases from 0 to \( +\infty \).
- From \( \theta = \frac{\pi}{2} \) to \( \theta = \pi \), \( \tan \theta \) decreases from \( -\infty \) to 0.
- From \( \theta = \pi \) to \( \theta = \frac{3\pi}{2} \), \( \tan \theta \) increases from 0 to \( +\infty \. 3. **Locating the Value -2.747**: Since \( \tan \theta = -2.747 \) is negative, we need to find the interval where tangent values are negative:
- The tangent function is negative in the second quadrant (from \( \frac{\pi}{2} \) to \( \pi \)) and the fourth quadrant (from \( \frac{3\pi}{2} \) to \( 2\pi \)).

4. **Finding the Specific Interval**:
- In the second quadrant, \( \tan \theta \) starts from \( -\infty \) at \( \frac{\pi}{2} \) and approaches 0 at \( \pi \). Therefore, \( -2.747 \) must lie in the interval \( \left(\frac{\pi}{2}, \pi\right) \).
- In the fourth quadrant, \( \tan \theta \) starts from 0 at \( 2\pi \) and approaches \( -\infty \) at \( \frac{3\pi}{2} \). However, since we are looking for values less than \( -1 \), we focus on the second quadrant.

5. **Conclusion**:
The value \( -2.747 \) lies between \( -\infty \) and 0, specifically in the interval \( \left(\frac{\pi}{2}, \pi\right) \). Thus, the possible values of \( \theta \) that satisfy \( \tan \theta \approx -2.747 \) are within this interval.

### Final Answer:
The possible value of \( \theta \) is in the interval \( \left(\frac{\pi}{2}, \pi\right) \).

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