Question

Let f(x) be a polynomial such that f(cos θ) = cos(4) θ for all θ. Find f(x). (This is essentially the same as finding cos(4) θ in terms of cos θ we structure the problem this way so that you can answer as a polynomial. Be sure to write your polynomial with the terms in order of decreasing degree.)

Answer 1

f(x) = 8x⁴-8x²+1

Explanation:

I will assume that f(cos θ) = cos(4θ). Otherwise, f would not be a polynomial. lets divide cos(4θ) in an expression depending on cos(θ). We use this properties

• cos(2a) = cos²(a) - sin²(b)
• sin(2a) = 2sin(a)cos(a)
• sin²(a) = 1-cos²(a)

cos(4θ) = cos(2 * (2θ) ) = cos²(2θ) - sin²(2θ) = [ cos²(θ)-sin²(θ) ]² - [2cos(θ)sin(θ)]² = [cos²(θ) - ( 1 - cos²(θ) ) ]² - 4cos²(θ)sin²(θ) = [2cos²(θ)-1]² - 4cos²(θ) (1 - cos²(θ) ) = 4 cos⁴(θ) - 4 cos²(θ) + 1 - 4 cos²(θ) + 4 cos⁴(θ) = 8cos⁴(θ) - 8 cos²(θ) + 1

Thus f(cos(θ)) = 8 cos⁴(θ) - 8 cos²(θ) + 1, and, as a result

f(x) = 8x⁴-8x²+1.