Solve for x tan(x)=2sin(x)

Problem

tan(x)=2*sin(x)

Solution

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

sin(x)/cos(x)=2*sin(x)

2. Subtract 2*sin(x) from both sides to set the equation to zero.

sin(x)/cos(x)−2*sin(x)=0

3. Factor out the common term sin(x)

sin(x)*(1/cos(x)−2)=0

4. Set each factor to zero to find the possible solutions for x

sin(x)=0

1/cos(x)−2=0

5. Solve the first equation sin(x)=0 which occurs at integer multiples of π

x=n*π

6. Solve the second equation by isolating cos(x)

1/cos(x)=2

cos(x)=1/2

7. Determine the values of x where cos(x)=1/2 within one period.

x=π/3+2*n*π

x=−π/3+2*n*π

Final Answer

tan(x)=2*sin(x)⇒x=n*π,±π/3+2*n*π

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