Solve for x 3cos(2x)-5cos(x)=1

Problem

3*cos(2*x)−5*cos(x)=1

Solution

1. Apply the double angle identity for cos(2*x) to rewrite the equation in terms of cos(x) only, using cos(2*x)=2*cos2(x)−1

3*(2*cos2(x)−1)−5*cos(x)=1

2. Distribute the constant 3 into the parentheses.

6*cos2(x)−3−5*cos(x)=1

3. Rearrange the equation into standard quadratic form by subtracting 1 from both sides.

6*cos2(x)−5*cos(x)−4=0

4. Factor the quadratic expression, treating cos(x) as the variable.

(3*cos(x)−4)*(2*cos(x)+1)=0

5. Set each factor to zero to solve for cos(x)

3*cos(x)−4=0⇒cos(x)=4/3

2*cos(x)+1=0⇒cos(x)=−1/2

6. Evaluate the solutions. Since the range of cos(x) is [−1,1] the equation cos(x)=4/3 has no real solutions.

7. Solve for x using the valid equation cos(x)=−1/2 within the general solution format, where n is any integer.

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

x=(2*π)/3+2*n*π,(4*π)/3+2*n*π

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