There is a clown's face on the top of a spinner. The tip of his hat rotates to #(-2, 5)# during one spin. What is the cosine value of this function?

To find the cosine value of the function representing the spinner's motion, we need to determine the horizontal displacement (change in x-coordinate) of the tip of the clown's hat.

Given that the tip of the hat rotates to (-2, 5) during one spin, the initial position of the tip of the hat is at (0, 0), since it starts at the center of rotation.

The horizontal displacement, or change in x-coordinate, is the difference between the x-coordinate of the final position and the x-coordinate of the initial position:

( \Delta x = -2 - 0 = -2 )

The cosine value of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. In this case, the hypotenuse is the radius of the spinner, which is the distance from the center of rotation to the tip of the hat.

Using the Pythagorean theorem, we can find the length of the radius (hypotenuse):

( r = \sqrt{(-2 - 0)^2 + (5 - 0)^2} = \sqrt{4 + 25} = \sqrt{29} )

Now, we can calculate the cosine value using the adjacent side (change in x-coordinate) and the hypotenuse:

( \cos(\theta) = \frac{\Delta x}{r} = \frac{-2}{\sqrt{29}} )

Therefore, the cosine value of the function representing the spinner's motion is ( \frac{-2}{\sqrt{29}} ).