Evaluate: tan-1(1) + cos-1(−1/2) + sin-1(−1/2)
Solution:
Using principal values:
\[ \tan^{-1}(1) = \frac{\pi}{4} \]
\[ \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \]
(Using standard principal value ranges of inverse trigonometric functions :contentReference[oaicite:0]{index=0})
Now,
\[ \frac{\pi}{4} + \frac{2\pi}{3} – \frac{\pi}{6} \]
Take LCM = 12:
\[ = \frac{3\pi}{12} + \frac{8\pi}{12} – \frac{2\pi}{12} = \frac{9\pi}{12} = \frac{3\pi}{4} \]
Final Answer:
Value = \[ \frac{3\pi}{4} \]