Final answer:
To find cos(y) for y=arcsin(x), consider a right-angled triangle where the hypotenuse is 1 and the opposite side to angle y is x, then use the Pythagorean theorem to find the adjacent side. The cosine of y equals the adjacent side length, which is the square root of (1 - x^2).
Step-by-step explanation:
To find cos(y) when given y=arcsin(x), where 0 is given for the range of y, we can reference the properties of right-angled triangles and trigonometric identities. Given that y is an angle for which the sine is x, by the definition of the arcsine function, we can consider a right-angled triangle where the opposite side to angle y has a length of x and the hypotenuse has a length of 1 (since the sine of y is x/1). Based on Pythagorean theorem, we have the adjacent side length, which we can call 'a', fulfills the equation a^2 + x^2 = 1^2. Solving for a gives us a = √(1-x^2). Since cosine is the ratio of the adjacent side over the hypotenuse, cos(y) is equal to 'a', thus cos(y) = √(1-x^2).