Solve for x 4cos(x)^2=3

Problem

4*cos2(x)=3

Solution

1. Isolate the squared trigonometric term by dividing both sides of the equation by 4

cos2(x)=3/4

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

cos(x)=±√(,3/4)

3. Simplify the radical expression on the right side.

cos(x)=±√(,3)/2

4. Identify the reference angle. The cosine of an angle is √(,3)/2 when the angle is π/6 (or 30.

(x_ref)=π/6

5. Determine the solutions within one period [0,2*π) Since the cosine is both positive and negative, solutions exist in all four quadrants.

x=π/6,(5*π)/6,(7*π)/6,(11*π)/6

6. Generalize the solution by adding multiples of the period. Since these values are separated by exactly π (e.g., π/6 and (7*π)/6, the solution can be condensed.

x=π/6+n*π

x=(5*π)/6+n*π

Final Answer

x=π/6+n*π,(5*π)/6+n*π

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