Compute inverse functions to four significant digits.

cos^2x=3-5cosx

rewrite it as..

cos^2x + 5cosx -3=0

now you have a quadratic, solve for cos x using the equation

cosx=(-5 -+ sqrt (25 +12)/2

To find the inverse function, let's first solve for cos(x) in the quadratic equation cos^2x + 5cosx - 3 = 0.

Using the quadratic formula:

cos(x) = (-5 ± √(25 + 12)) / 2

cos(x) = (-5 ± √37) / 2

Now, to find the inverse function, we need to swap the x and y variables and solve for y:

y = (-5 ± √37) / 2

Therefore, the inverse functions are:

cos^(-1)((-5 + √37) / 2) and cos^(-1)((-5 - √37) / 2)

These inverse functions give the input value of x when the output value of cos(x) is equal to (-5 + √37) / 2 or (-5 - √37) / 2, respectively.