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.

1/√(,2)*sin(x)+1/√(,2)*cos(x)=1

2. Substitute the values cos(45^∘)=1/√(,2) and sin(45^∘)=1/√(,2) (or π/4 radians) into the equation.

sin(x)*cos(π/4)+cos(x)*sin(π/4)=1

3. Apply the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

sin(x+π/4)=1

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

x+π/4=π/2+2*n*π

5. Isolate x by subtracting π/4 from both sides.

x=π/2−π/4+2*n*π

6. Simplify the constant terms.

x=π/4+2*n*π

Final Answer

sin(x)+cos(x)=√(,2)⇒x=π/4+2*n*π,n∈ℤ

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