Solve for x 4cos(x)^2=1

Problem

4*cos^2(x)=1

Solution

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

cos^2(x)=1/4

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

cos(x)=±√(,1/4)

3. Simplify the square root on the right side of the equation.

cos(x)=±1/2

4. Identify the reference angle where the cosine is 1/2 which is π/3 (or 60^∘.

x=π/3

5. Determine all solutions within one period [0,2*π) by finding where cos(x)=1/2 (Quadrants I and IV) and cos(x)=−1/2 (Quadrants II and III).

x=π/3,(2*π)/3,(4*π)/3,(5*π)/3

6. Generalize the solution by adding multiples of the period π since these values are spaced exactly π apart in pairs.

x=π/3+n*π

x=(2*π)/3+n*π

Final Answer

x=π/3+n*π,(2*π)/3+n*π

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