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

Problem

cos(x)=1/2

Solution

1. Take the square root of both sides of the equation to isolate the cosine function.

cos(x)=±√(,1/2)

2. Simplify the radical by rationalizing the denominator.

cos(x)=±1/√(,2)

cos(x)=±√(,2)/2

3. Identify the reference angle for which the cosine value is √(,2)/2

(x_ref)=π/4

4. Determine the general solutions for both the positive and negative cases. Since cos(x)=√(,2)/2 or cos(x)=−√(,2)/2 the solutions occur in all four quadrants at odd multiples of π/4

x=π/4+(n*π)/2

Final Answer

cos(x)=1/2⇒x=π/4+(n*π)/2,n∈ℤ

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