Evaluate tan(½ sin⁻¹(3/4))

Evaluate \( \tan\left(\frac{1}{2}\sin^{-1}\left(\frac{3}{4}\right)\right) \)

Solution:

Let

\[ \theta = \sin^{-1}\left(\frac{3}{4}\right) \]

Then,

\[ \sin \theta = \frac{3}{4} \]

Consider a right triangle:

So,

\[ \cos \theta = \sqrt{1 – \sin^2\theta} = \frac{\sqrt{7}}{4} \]

Using half-angle identity:

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

\[ = \frac{1 – \frac{\sqrt{7}}{4}}{\frac{3}{4}} \]

\[ = \frac{\frac{4 – \sqrt{7}}{4}}{\frac{3}{4}} \]

\[ = \frac{4 – \sqrt{7}}{3} \]

\[ \tan\left(\frac{1}{2}\sin^{-1}\left(\frac{3}{4}\right)\right) = \frac{4 – \sqrt{7}}{3} \]

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