write the expression as a function of x, with no angle measure involved\n\\( \\sin \\left( \\frac { \\pi } {…

write the expression as a function of x, with no angle measure involved\n\\( \\sin \\left( \\frac { \\pi } { 4 } + x \\right) \\)\n\\( \\sin \\left( \\frac { \\pi } { 4 } + x \\right) = \\square \\)\n(simplify your answer, including any radicals. use integers or fractions for

write the expression as a function of x, with no angle measure involved\n\\( \\sin \\left( \\frac { \\pi } { 4 } + x \\right) \\)\n\\( \\sin \\left( \\frac { \\pi } { 4 } + x \\right) = \\square \\)\n(simplify your answer, including any radicals. use integers or fractions for

Answer

Explanation:

Step1: Use the sine addition formula

The sine addition formula is $\sin(A + B)=\sin A\cos B+\cos A\sin B$. Here $A=\frac{\pi}{4}$ and $B = x$. So, $\sin(\frac{\pi}{4}+x)=\sin\frac{\pi}{4}\cos x+\cos\frac{\pi}{4}\sin x$.

Step2: Substitute the values of $\sin\frac{\pi}{4}$ and $\cos\frac{\pi}{4}$

We know that $\sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. Substituting these values, we get $\frac{\sqrt{2}}{2}\cos x+\frac{\sqrt{2}}{2}\sin x$.

Step3: Factor out the common term

Factor out $\frac{\sqrt{2}}{2}$, so the expression becomes $\frac{\sqrt{2}}{2}(\sin x+\cos x)$.

Answer:

$\frac{\sqrt{2}}{2}(\sin x+\cos x)$