Let f(x) be a polynomial such that f(cos theta) = cos(4 theta) for all theta. Find f(x). (This is essentially the same as finding cos(4 theta) in terms of cos theta; 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.)

Question

Let f(x) be a polynomial such that f(cos theta) = cos(4 theta) for all theta. Find f(x). (This is essentially the same as finding cos(4 theta) in terms of cos theta; 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.)

Note: The answer is NOT 8cos^4x+8cos^2x+1

Answer 1

cos 4Ø

= 2cos^2 2Ø - 1)
= 2(2cos^2 Ø - 1)(2cos^2 Ø -1) -1
= 2(4cos^4 Ø - 4cos^2 Ø + 1) - 1
= 8cos^4 Ø - 8cos^2 Ø + 2 - 1
= 8cos^4 Ø - 8cos^2 Ø + 1

now let cosØ = x and we have

f(x) = 8x^4 - 8x^2 + 1

Answer 2

OHHH it has to be in terms of x! Thanks!

Answer 3

To find the polynomial f(x), we can expand the right side of the equation and compare the terms with the left side.

Using the double-angle formula for cosine, we have:

cos(4theta) = cos^2(2theta) - sin^2(2theta)
= (cos^2(theta) - sin^2(theta))^2 - (2sin(theta)cos(theta))^2
= cos^4(theta) - 2cos^2(theta)sin^2(theta) + sin^4(theta) - 4sin^2(theta)cos^2(theta)
= cos^4(theta) + sin^4(theta) - 6sin^2(theta)cos^2(theta)

Now let's replace theta with x and cos(theta) with x in the equation:

f(x) = cos^4(x) + sin^4(x) - 6sin^2(x)cos^2(x)

Since cos^2(x) + sin^2(x) = 1, we can substitute 1 - sin^2(x) for cos^2(x):

f(x) = (1 - sin^2(x))^2 + sin^4(x) - 6sin^2(x)(1 - sin^2(x))
= 1 - 2sin^2(x) + sin^4(x) + sin^4(x) - 6sin^2(x) + 6sin^4(x)
= 8sin^4(x) - 8sin^2(x) + 1

Therefore, the polynomial f(x) = 8x^4 - 8x^2 + 1.

Answer 4

To find the polynomial f(x) such that f(cos(theta)) = cos(4theta) for all theta, we can begin by expanding cos(4theta) using the multiple angle formula. The multiple angle formula for cosine states that:

cos(4theta) = cos^4(theta) - 6cos^2(theta)sin^2(theta) + sin^4(theta)

Now that we have cos(4theta) written in terms of cos(theta) and sin(theta), we need to express sin^2(theta) in terms of cos^2(theta) to eliminate the sine terms. Recall the trigonometric identity:

sin^2(theta) = 1 - cos^2(theta)

Now, substitute sin^2(theta) with 1 - cos^2(theta) in the expanded form of cos(4theta):

cos(4theta) = cos^4(theta) - 6cos^2(theta)(1 - cos^2(theta)) + (1 - cos^2(theta))^2

Simplifying further:

cos(4theta) = cos^4(theta) - 6cos^2(theta) + 6cos^4(theta) + 1 - 2cos^2(theta) + cos^4(theta)

Combining like terms:

cos(4theta) = 8cos^4(theta) - 8cos^2(theta) + 1

Comparing this with f(cos(theta)), we can conclude that:

f(x) = 8x^4 - 8x^2 + 1

Therefore, the polynomial f(x) in terms of x is given by 8x^4 - 8x^2 + 1.