Use the Rational Roots Test to Find All Possible Roots cos(2x)

Problem

cos(2*x)

Solution

1. Identify the expression as a trigonometric function. The Rational Roots Test is a theorem used to find potential rational roots of a polynomial equation with integer coefficients.

2. Determine if the test applies. The expression cos(2*x) is not a polynomial in terms of x Therefore, the Rational Roots Test cannot be directly applied to the variable x in this expression.

3. Rewrite the expression using a trigonometric identity if a polynomial form is desired. Using the double-angle identity, we have cos(2*x)=2*cos2(x)−1

4. Define a polynomial by substitution. Let u=cos(x) The expression becomes the polynomial P(u)=2*u2−1

5. Apply the Rational Roots Test to 2*u2−1=0 The constant term is (a_0)=−1 and the leading coefficient is (a_n)=2

6. List the factors of the constant term p=±1 and the factors of the leading coefficient q=±1,±2

7. Calculate the possible rational roots for u by finding all possible values of p/q These are ±1 and ±1/2

Final Answer

Possible Rational Roots for *cos(x)=±1,±1/2

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