Solve for x sin(2x)-cos(x)=0

Problem

sin(2*x)−cos(x)=0

Solution

1. Apply the double angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x)

sin(2*x)−cos(x)=0

2*sin(x)*cos(x)−cos(x)=0

2. Factor out the common term cos(x) from the expression.

cos(x)*(2*sin(x)−1)=0

3. Set each factor to zero using the zero product property.

cos(x)=0

2*sin(x)−1=0

4. Solve the first equation for x within the standard interval [0,2*π)

cos(x)=0

x=π/2,(3*π)/2

5. Solve the second equation for x by isolating sin(x)

2*sin(x)=1

sin(x)=1/2

x=π/6,(5*π)/6

6. Generalize the solutions by adding multiples of the period 2*π*n where n is an integer. Note that π/2 and (3*π)/2 can be combined into π/2+π*n

x=π/2+π*n

x=π/6+2*π*n

x=(5*π)/6+2*π*n

Final Answer

sin(2*x)−cos(x)=0⇒x=π/2+π*n,π/6+2*π*n,(5*π)/6+2*π*n

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