Evaluate sin(½ cos⁻¹(4/5))

Evaluate \( \sin\left(\frac{1}{2}\cos^{-1}\left(\frac{4}{5}\right)\right) \)

Solution:

Let

\[ \theta = \cos^{-1}\left(\frac{4}{5}\right) \]

Then,

\[ \cos \theta = \frac{4}{5} \]

Using identity:

\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 – \cos\theta}{2}} \]

\[ = \sqrt{\frac{1 – \frac{4}{5}}{2}} \]

\[ = \sqrt{\frac{\frac{1}{5}}{2}} \]

\[ = \sqrt{\frac{1}{10}} \]

\[ = \frac{1}{\sqrt{10}} \]

\[ \sin\left(\frac{1}{2}\cos^{-1}\left(\frac{4}{5}\right)\right) = \frac{1}{\sqrt{10}} \]

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