Solve for x 2cos(x)^2-11cos(x)+5=0

Problem

2*cos2(x)−11*cos(x)+5=0

Solution

1. Substitute a variable to simplify the quadratic form by letting u=cos(x)

2*u2−11*u+5=0

2. Factor the quadratic equation by finding two numbers that multiply to 2⋅5=10 and add to −11

(2*u−1)*(u−5)=0

3. Solve for u by setting each factor to zero.

2*u−1=0⇒u=1/2

u−5=0⇒u=5

4. Back-substitute u=cos(x) into the results.

cos(x)=1/2

cos(x)=5

5. Evaluate the validity of the solutions. Since the range of the cosine function is [−1,1] the equation cos(x)=5 has no real solutions.

cos(x)=5⇒No solution

6. Find the general solutions for cos(x)=1/2 using the unit circle.

x=π/3+2*n*π

x=−π/3+2*n*π

Final Answer

x=±π/3+2*n*π

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