Question
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,
\[ \cos \theta = \frac{4}{5} \]
Construct a right triangle:
- Adjacent = 4
- Hypotenuse = 5
- Opposite = 3
So,
\[ \sin \theta = \frac{3}{5} \]
Now use half-angle identity:
\[ \sin\frac{\theta}{2} = \sqrt{\frac{1 – \cos \theta}{2}} \]
Substitute:
\[ \sin\frac{\theta}{2} = \sqrt{\frac{1 – \frac{4}{5}}{2}} \]
\[ = \sqrt{\frac{\frac{1}{5}}{2}} = \sqrt{\frac{1}{10}} \]
\[ = \frac{1}{\sqrt{10}} \]
Final Answer:
\[ \boxed{\frac{1}{\sqrt{10}}} \]
Key Concept
Use triangle representation and half-angle identity to simplify inverse trigonometric expressions.