Solve for x 2cos(x)^2+cos(x)-1=0

Problem

2*cos2(x)+cos(x)−1=0

Solution

1. Identify the equation as a quadratic in terms of cos(x)

2. Substitute u=cos(x) to rewrite the equation as a standard quadratic equation.

2*u2+u−1=0

3. Factor the quadratic expression by finding two numbers that multiply to −2 and add to 1

(2*u−1)*(u+1)=0

4. Solve for u by setting each factor equal to zero.

2*u−1=0⇒u=1/2

u+1=0⇒u=−1

5. Back-substitute cos(x) for u to find the values of x

cos(x)=1/2

cos(x)=−1

6. Determine the general solutions for x using the properties of the cosine function.

x=π/3+2*n*π

x=(5*π)/3+2*n*π

x=π+2*n*π

7. Combine the solutions where n is any integer.

Final Answer

x=±π/3+2*n*π,π+2*n*π

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