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) to rewrite the equation.

sin(2*x)+cos(x)=0

2*sin(x)*cos(x)+cos(x)=0

2. Factor out the common term cos(x) from the left side of the equation.

cos(x)*(2*sin(x)+1)=0

3. Set each factor to zero using the zero product property to find the possible values for x

cos(x)=0

2*sin(x)+1=0

4. Solve the first equation cos(x)=0 The cosine function is zero at odd multiples of π/2

x=π/2+n*π

5. Solve the second equation 2*sin(x)+1=0 by isolating sin(x)

2*sin(x)=−1

sin(x)=−1/2

6. Determine the angles where sin(x)=−1/2 In the interval [0,2*π) these occur at (7*π)/6 and (11*π)/6

x=(7*π)/6+2*n*π

x=(11*π)/6+2*n*π

Final Answer

sin(2*x)+cos(x)=0⇒x=π/2+n*π,(7*π)/6+2*n*π,(11*π)/6+2*n*π

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