Question

unit test
unit test review active
which shows a correct simplification of sin(x + π)?
sin(x + π)
= sin(x)cos(π) + cos(x)sin(π)
= sin(x) · 1 + cos(x) · 0
= sin(x)
sin(x + π)
= cos(x)cos(π) - sin(x)sin(π)
= cos(x) · 1 - sin(x) · 0
= cos(x)
sin(x + π)
= sin(x)cos(π) + cos(x)sin(π)
= sin(x) · -1 + cos(x) · 0
= -sin(x)
sin(x + π)
= cos(x)cos(π) - sin(x)sin(π)
= cos(x) · -1 - sin(x) · 0
= -cos(x)

Explanation:

Step1: Recall the sine addition formula

The formula for \(\sin(A + B)=\sin(A)\cos(B)+\cos(A)\sin(B)\). Here \(A = x\) and \(B=\pi\).

Step2: Substitute the values of \(\cos(\pi)\) and \(\sin(\pi)\)

We know that \(\cos(\pi)=- 1\) and \(\sin(\pi)=0\).
Substitute into \(\sin(x+\pi)=\sin(x)\cos(\pi)+\cos(x)\sin(\pi)\)

$$
LATEXBLOCK0
$$

Answer:

The correct simplification is \(\sin(x+\pi)=\sin(x)\cos(\pi)+\cos(x)\sin(\pi)=-\sin(x)\) (assuming the option that follows this calculation is correct among the given choices in the original multiple - choice setup).