Simplify sin(pi/4+x)

Problem

sin(π/4+x)

Solution

1. Identify the sum formula for the sine function, which is sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

2. Substitute the values A=π/4 and B=x into the formula.

sin(π/4+x)=sin(π/4)*cos(x)+cos(π/4)*sin(x)

3. Evaluate the trigonometric constants using the unit circle, where sin(π/4)=√(,2)/2 and cos(π/4)=√(,2)/2

sin(π/4+x)=√(,2)/2*cos(x)+√(,2)/2*sin(x)

4. Factor out the common term √(,2)/2 to simplify the expression.

sin(π/4+x)=√(,2)/2*(cos(x)+sin(x))

Final Answer

sin(π/4+x)=√(,2)/2*(sin(x)+cos(x))

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