Solve for x tan(x)=(2 square root of 3)/3*sin(x)

Problem

tan(x)=(2√(,3))/3*sin(x)

Solution

1. Rewrite the tangent function using the identity tan(x)=sin(x)/cos(x)

sin(x)/cos(x)=(2√(,3))/3*sin(x)

2. Rearrange the equation by moving all terms to one side to set it equal to zero.

sin(x)/cos(x)−(2√(,3))/3*sin(x)=0

3. Factor out the common term sin(x) from the expression.

sin(x)*(1/cos(x)−(2√(,3))/3)=0

4. Set the first factor to zero to find the first set of solutions.

sin(x)=0

x=n*π

5. Set the second factor to zero to find the remaining solutions.

1/cos(x)−(2√(,3))/3=0

1/cos(x)=(2√(,3))/3

6. Invert both sides to solve for cos(x)

cos(x)=3/(2√(,3))

7. Rationalize the denominator of the fraction.

cos(x)=(3√(,3))/(2⋅3)

cos(x)=√(,3)/2

8. Determine the values of x that satisfy the cosine equation.

x=π/6+2*n*π

x=(11*π)/6+2*n*π

Final Answer

tan(x)=(2√(,3))/3*sin(x)⇒x=n*π,π/6+2*n*π,(11*π)/6+2*n*π

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