Question
If
\[ \alpha = \tan^{-1}(\tan \tfrac{5\pi}{4}), \quad \beta = \tan^{-1}(-\tan \tfrac{2\pi}{3}) \]
Find the relation between \( \alpha \) and \( \beta \).
Solution
Principal value range of \( \tan^{-1}x \) is:
\[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]
Find α:
\[ \tan \tfrac{5\pi}{4} = 1 \Rightarrow \alpha = \tan^{-1}(1) = \frac{\pi}{4} \]
Find β:
\[ -\tan \tfrac{2\pi}{3} = \sqrt{3} \Rightarrow \beta = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]
Relation:
\[ \alpha = \frac{\pi}{4}, \quad \beta = \frac{\pi}{3} \]
\[ 4\alpha = 4 \cdot \frac{\pi}{4} = \pi \]
\[ 3\beta = 3 \cdot \frac{\pi}{3} = \pi \]
\[ \therefore \; 4\alpha = 3\beta \]
Final Answer:
\[ \boxed{4\alpha = 3\beta} \]
Key Concept
Convert both angles into numerical values and compare multiples to get the relation.