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 among \( \tan(-4) \), \( \tan(-7) \), \( \tan(-10) \), and \( \tan(-1) \), we can use the periodic properties of the tangent function and its behavior in different quadrants.

### Step 1: Use the periodicity of the tangent function
The tangent function has a period of \( \pi \). This means:
\[ \tan(x + n\pi) = \tan(x) \quad \text{for any integer } n \]
We can use this property to simplify our angles.

### Step 2: Convert angles to a standard range
We will convert each angle to a standard range (usually between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \)):
- For \( \tan(-4) \):
\[ -4 + 5\pi \quad (\text{since } 5\pi \approx 15.7 \text{ is greater than } 4) \]
\[ \tan(-4) = \tan(5\pi - 4) = \tan(1.283) \approx 0.86 \]

- For \( \tan(-7) \):
\[ -7 + 8\pi \quad (\text{since } 8\pi \approx 25.13 \text{ is greater than } 7) \]
\[ \tan(-7) = \tan(8\pi - 7) = \tan(1.57) \approx 0.72 \]

- For \( \tan(-10) \):
\[ -10 + 11\pi \quad (\text{since } 11\pi \approx 34.56 \text{ is greater than } 10) \]
\[ \tan(-10) = \tan(11\pi - 10) = \tan(1.57) \approx 0.58 \]

- For \( \tan(-1) \):
\[ \tan(-1) = -\tan(1) \approx -1.557 \]

### Step 3: Compare the values
Now we have the approximate values:
- \( \tan(-4) \approx 0.86 \)
- \( \tan(-7) \approx 0.72 \)
- \( \tan(-10) \approx 0.58 \)
- \( \tan(-1) \approx -1.557 \)

### Step 4: Determine the greatest value
From the calculated values, we can see:
\[ \tan(-4) > \tan(-7) > \tan(-10) > \tan(-1) \]
Thus, the greatest value among these is:
\[ \tan(-4) \]

### Conclusion
The greatest among \( \tan(-4) \), \( \tan(-7) \), \( \tan(-10) \), and \( \tan(-1) \) is \( \tan(-4) \).