Question

Find the approximate value of \( \tan^{-1} (1.002) \). \[ {Given: } \pi = 3.1416. \]

Hint:

For small \( x \), use the approximation: \[ \tan^{-1} (1 + x) \approx \frac{\pi}{4} + x. \]

Answer 1

Step 1: Use Approximation Formula 
For small \( x \), \( \tan^{-1} (1 + x) \approx \frac{\pi}{4} + x \). 
Step 2: Apply Given Values 
\[ \tan^{-1} (1.002) \approx \frac{\pi}{4} + 0.002. \] \[ = \frac{3.1416}{4} + 0.002. \] \[ = 0.7854 + 0.002 = 0.7874. \]