Problem
Solve: \( \cos(\sin^{-1}x) = \frac{1}{6} \)
Solution
Let \( \theta = \sin^{-1}x \)
Then:
\[ \sin \theta = x = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]
So take:
- Perpendicular = x
- Hypotenuse = 1
Base:
\[ \sqrt{1 – x^2} \]
Now,
\[ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \sqrt{1 – x^2} \]
Given:
\[ \sqrt{1 – x^2} = \frac{1}{6} \]
Squaring both sides:
\[ 1 – x^2 = \frac{1}{36} \]
\[ x^2 = \frac{35}{36} \]
\[ x = \pm \frac{\sqrt{35}}{6} \]
Final Answer
\[ \boxed{x = \pm \frac{\sqrt{35}}{6}} \]
Explanation
Using identity: cos(sin⁻¹x) = √(1 − x²), then solving the equation.