Question

Find the equivalent function of sin(180+x)

Answer 1

The equivalent function of
`sin\left(180^(\circ )+x\right)` will be:
`-\sin \left(x\right)`

i.e.

`\sin \left(180^(\circ \:)+x\right)=-\sin \left(x\right)`

Explanation:

Considering the function

`sin\left(180^(\circ )+x\right)`

Solving

`sin\left(180^(\circ )+x\right)`

Using the angle sum identity:

`\sin \left(s+t\right)=\sin \left(s\right)\cos \left(t\right)+\cos \left(s\right)\sin \left(t\right)`

`=\sin \left(180^(\circ \:)\right)\cos \left(x\right)+\cos \left(180^(\circ \:)\right)\sin \left(x\right)`

as

`\cos \left(180^(\circ \:)\right)=\left(-1\right)`

`\sin \left(180^(\circ \:)\right)=0`

so

`=0\cdot \:\cos \:\left(x\right)+\left(-1\right)\cdot \:\sin \:\left(x\right)`

`=0\cdot \cos \left(x\right)-1\cdot \sin \left(x\right)`

`=0-1\cdot \sin \left(x\right)`

`=0-\sin \left(x\right)`

`=-\sin \left(x\right)`

So, the equivalent function of
`sin\left(180^(\circ )+x\right)` will be:
`-\sin \left(x\right)`

Therefore:

`\sin \left(180^(\circ \:)+x\right)=-\sin \left(x\right)`