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. Recognize that 1/√(,2) is equal to both cos(45) and sin(45) or in radians, cos(π/4) and sin(π/4)

cos(π/4)*sin(x)+sin(π/4)*cos(x)=1

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

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

x=π/4+2*n*π

Final Answer

sin(x)+cos(x)=√(,2)⇒x=π/4+2*n*π

Want more problems? Check here!