Question

The principal value of \( \tan^{-1}(1) \) is:

• A. (0)
• B. (\(\frac{\pi}{6}\))
• C. (\(\frac{\pi}{4}\))
• D. (\(\frac{\pi}{2}\))

Hint:

Memorize the principal value ranges for inverse trigonometric functions:
- \( \sin^{-1}(x) \): \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
- \( \cos^{-1}(x) \): \([0, \pi]\)
- \( \tan^{-1}(x) \): \((-\frac{\pi}{2}, \frac{\pi}{2})\)
These ranges ensure a unique output for each input.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the principal value of the inverse tangent function for the input 1. The principal value refers to the unique value within the restricted range of the inverse trigonometric function.

Step 2: Key Formula or Approach:
The principal value branch for \( \tan^{-1}(x) \) is \((-\frac{\pi}{2}, \frac{\pi}{2})\) (exclusive of endpoints).
We need to find the angle \(\theta\) such that \( \tan(\theta) = 1 \) and \(\theta\) lies within this range.

Step 3: Detailed Explanation:
We are looking for an angle \(\theta\) such that \( \tan(\theta) = 1 \).
We know that \( \tan(45^{\circ}) = 1 \).
Converting $45^{\circ}$ to radians:
$45^{\circ} = 45 \times \frac{\pi}{180} \text{ radians} = \frac{\pi}{4} \text{ radians}$.
The value \( \frac{\pi}{4} \) lies within the principal value branch of \( \tan^{-1}(x) \), which is \((-\frac{\pi}{2}, \frac{\pi}{2})\).
Therefore, the principal value of \( \tan^{-1}(1) \) is \( \frac{\pi}{4} \).

Step 4: Final Answer:
The principal value is \(\frac{\pi}{4}\).