Write the expression in algebraic form. (Hint: Sketch a right triangle.) \[ \cos (\arcsin (5 x)) \]

To express \( \cos (\arcsin (5 x)) \) in algebraic form, consider a right triangle where the angle \( \theta = \arcsin(5x) \). Here, the opposite side is \( 5x \) and the hypotenuse is \( 1 \) (since \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)). By using the Pythagorean theorem, the adjacent side can be calculated as \( \sqrt{1^2 - (5x)^2} = \sqrt{1 - 25x^2} \). Thus, the cosine of the angle \( \theta \) can be expressed as: \[ \cos (\arcsin (5x)) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{1 - 25x^2}}{1} = \sqrt{1 - 25x^2}. \] Next, remember that \( 5x \) must stay within the range [-1, 1] for the arcsine function to be defined, so ensure \( -\frac{1}{5} \leq x \leq \frac{1}{5} \)!