Find the exact value of the inverse trig function expressions

Consider the expression $ \sin^{-1}\left( \frac{\sqrt{3}}{2} \right) $.

We know that $ \sin\left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} $.

Therefore, $ \sin^{-1}\left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{3} $.

Next, consider the expression $ \tan^{-1}(1) $.

We know that $ \tan\left( \frac{\pi}{4} \right) = 1 $.

Therefore, $ \tan^{-1}(1) = \frac{\pi}{4} $.

Finally, consider the expression $ \cos^{-1}\left( -\frac{1}{2} \right) $.

We know that $ \cos\left( \pi – \frac{\pi}{3} \right) = -\frac{1}{2} $.

Therefore, $ \cos^{-1}\left( -\frac{1}{2} \right) = \frac{2\pi}{3} $.

In summary:

$ \sin^{-1}\left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{3} $

$ \tan^{-1}(1) = \frac{\pi}{4} $

$ \cos^{-1}\left( -\frac{1}{2} \right) = \frac{2\pi}{3} $

Let

For $ sin^{-1}left( frac{sqrt{3}}{2}
ight) $,

it is $ frac{pi}{3} $.

For $ an^{-1}(1) $,

it is $ frac{pi}{4} $.

For $ cos^{-1}left( -frac{1}{2}
ight) $,

it is $ frac{2pi}{3} $.

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