Solve for x 4cos(x)^2-3=0

Problem

4*cos(x)−3=0

Solution

1. Isolate the squared trigonometric term by adding 3 to both sides and then dividing by 4

4*cos(x)=3

cos(x)=3/4

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

cos(x)=±√(,3/4)

cos(x)=±√(,3)/2

3. Identify the reference angle on the unit circle where the cosine value is √(,3)/2

x=π/6

4. Determine all solutions within one period [0,2*π) for both √(,3)/2 and −√(,3)/2

x=π/6,(5*π)/6,(7*π)/6,(11*π)/6

5. Generalize the solution by adding multiples of the period π since the solutions are spaced exactly π apart (or 2*π for each individual branch).

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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