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

Problem

sin(x)−cos(x)=√(,2)

Solution

1. Divide both sides of the equation by √(,2) to prepare for a trigonometric identity substitution.

1/√(,2)*sin(x)−1/√(,2)*cos(x)=1

2. Identify the values as sine and cosine of a known angle, specifically cos(π/4)=1/√(,2) and sin(π/4)=1/√(,2)

sin(x)*cos(π/4)−cos(x)*sin(π/4)=1

3. Apply the identity for the sine of a difference, sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B) where A=x and B=π/4

sin(x−π/4)=1

4. Solve for the argument by determining where the sine function equals 1

x−π/4=π/2+2*n*π

5. Isolate x by adding π/4 to both sides of the equation.

x=π/2+π/4+2*n*π

6. Simplify the constant terms to find the general solution.

x=(3*π)/4+2*n*π

Final Answer

sin(x)−cos(x)=√(,2)⇒x=(3*π)/4+2*n*π

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