Solve for x cos(2x)+6sin(x)^2=4

Problem

cos(2*x)+6*sin2(x)=4

Solution

1. Apply the double angle identity for cos(2*x) that involves sin2(x) which is cos(2*x)=1−2*sin2(x)

1−2*sin2(x)+6*sin2(x)=4

2. Combine like terms by adding the coefficients of the sin2(x) terms.

1+4*sin2(x)=4

3. Isolate the term containing the sine function by subtracting 1 from both sides.

4*sin2(x)=3

4. Divide by the coefficient of the sine term to solve for sin2(x)

sin2(x)=3/4

5. Take the square root of both sides, remembering to include both the positive and negative roots.

sin(x)=±√(,3)/2

6. Determine the reference angle and find all solutions within the standard interval [0,2*π)

x=π/3,(2*π)/3,(4*π)/3,(5*π)/3

7. Generalize the solution by adding multiples of π since the values are separated by π/3 and (2*π)/3 intervals.

x=π/3+π*k

x=(2*π)/3+π*k

Final Answer

cos(2*x)+6*sin2(x)=4⇒x=π/3+π*k,(2*π)/3+π*k

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