Which of the following is the greatest?

Options

A. tan 1
B. tan 4
C. tan 7
D. tan 10

Explanation

To determine which of the following values is the greatest: \( \tan(1) \), \( \tan(4) \), \( \tan(7) \), or \( \tan(10) \), we will follow these steps:

### Step 1: Understand the behavior of the tangent function
The tangent function, \( \tan(x) \), is periodic with a period of \( \pi \) (approximately \( 3.14 \)). It is also an increasing function in the intervals \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) and \( (\frac{\pi}{2}, \frac{3\pi}{2}) \).

### Step 2: Identify the quadrant for each angle
- \( 1 \) is in the first quadrant (since \( 1 < \pi \)).
- \( 4 \) is in the second quadrant (since \( \pi < 4 < 2\pi \)).
- \( 7 \) is in the third quadrant (since \( 2\pi < 7 < 3\pi \)).
- \( 10 \) is in the fourth quadrant (since \( 3\pi < 10 < 4\pi \)).

### Step 3: Convert angles to the first quadrant
To compare these values, we can convert each angle to an equivalent angle in the first quadrant.

1. **For \( \tan(4) \)**:
\[ \tan(4) = \tan(4 - \pi) = \tan(4 - 3.14) = \tan(0.86) \]

2. **For \( \tan(7) \)**:
\[ \tan(7) = \tan(7 - 2\pi) = \tan(7 - 6.28) = \tan(0.72) \]

3. **For \( \tan(10) \)**:
\[ \tan(10) = \tan(10 - 3\pi) = \tan(10 - 9.42) = \tan(0.58) \]

### Step 4: List the values to compare
Now we have:
- \( \tan(1) \)
- \( \tan(0.86) \) (from \( \tan(4) \))
- \( \tan(0.72) \) (from \( \tan(7) \))
- \( \tan(0.58) \) (from \( \tan(10) \))

### Step 5: Compare the values
Since \( \tan(x) \) is an increasing function in the first quadrant:
- \( 1 > 0.86 > 0.72 > 0.58 \)

Thus, we can conclude:
\[ \tan(1) > \tan(0.86) > \tan(0.72) > \tan(0.58) \]

### Final Conclusion
The greatest value among \( \tan(1) \), \( \tan(4) \), \( \tan(7) \), and \( \tan(10) \) is:
\[ \boxed{\tan(1)} \]