Answer:
The equivalent function of [tex]sin\left(180^{\circ }+x\right)[/tex] will be: [tex]-\sin \left(x\right)[/tex]
i.e.
[tex]\sin \left(180^{\circ \:}+x\right)=-\sin \left(x\right)[/tex]
Step-by-step explanation:
Considering the function
[tex]sin\left(180^{\circ }+x\right)[/tex]
Solving
[tex]sin\left(180^{\circ }+x\right)[/tex]
Using the angle sum identity:
[tex]\sin \left(s+t\right)=\sin \left(s\right)\cos \left(t\right)+\cos \left(s\right)\sin \left(t\right)[/tex]
[tex]=\sin \left(180^{\circ \:}\right)\cos \left(x\right)+\cos \left(180^{\circ \:}\right)\sin \left(x\right)[/tex]
as
[tex]\cos \left(180^{\circ \:}\right)=\left(-1\right)[/tex]
[tex]\sin \left(180^{\circ \:}\right)=0[/tex]
so
[tex]=0\cdot \:\cos \:\left(x\right)+\left(-1\right)\cdot \:\sin \:\left(x\right)[/tex]
[tex]=0\cdot \cos \left(x\right)-1\cdot \sin \left(x\right)[/tex]
[tex]=0-1\cdot \sin \left(x\right)[/tex]
[tex]=0-\sin \left(x\right)[/tex]
[tex]=-\sin \left(x\right)[/tex]
So, the equivalent function of [tex]sin\left(180^{\circ }+x\right)[/tex] will be: [tex]-\sin \left(x\right)[/tex]
Therefore:
[tex]\sin \left(180^{\circ \:}+x\right)=-\sin \left(x\right)[/tex]
