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

Problem

3*sin2(x)=cos2(x)

Solution

1. Divide both sides of the equation by cos2(x) to express the equation in terms of the tangent function, assuming cos(x)≠0

(3*sin2(x))/cos2(x)=cos2(x)/cos2(x)

2. Simplify the expression using the trigonometric identity tan(x)=sin(x)/cos(x)

3*tan2(x)=1

3. Isolate the squared tangent term by dividing both sides by 3

tan2(x)=1/3

4. Take the square root of both sides to solve for tan(x) remembering to include both the positive and negative roots.

tan(x)=±√(,1/3)

5. Rationalize the denominator of the constant.

tan(x)=±1/√(,3)

6. Find the reference angle for which tan(x)=1/√(,3) which is x=π/6 (or 30.

x=π/6+n*π

x=−π/6+n*π

7. Combine the solutions into a single general expression, where n is any integer.

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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