Solve for x cos(2x)+ square root of 2sin(x)=1

Problem

cos(2*x)+√(,2)*sin(x)=1

Solution

1. Apply the double angle identity for cosine to rewrite the equation in terms of sine.

cos(2*x)=1−2*sin2(x)

2. Substitute the identity into the original equation.

1−2*sin2(x)+√(,2)*sin(x)=1

3. Subtract 1 from both sides to simplify the equation.

−2*sin2(x)+√(,2)*sin(x)=0

4. Factor out the common term sin(x)

sin(x)*(−2*sin(x)+√(,2))=0

5. Set each factor to zero to find the possible values for sin(x)

sin(x)=0

−2*sin(x)+√(,2)=0

6. Solve for x in the first case where sin(x)=0

x=n*π

7. Solve for x in the second case by isolating sin(x)

−2*sin(x)=−√(,2)

sin(x)=√(,2)/2

8. Determine the angles where sin(x)=√(,2)/2

x=π/4+2*n*π

x=(3*π)/4+2*n*π

Final Answer

cos(2*x)+√(,2)*sin(x)=1⇒x=n*π,π/4+2*n*π,(3*π)/4+2*n*π

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