How would you show that sin(x+pi)=-sin x?

Question

I’m not sure how to start so if someone can tell me how to start it I should be able to do the rest.

Answer 1

construct a right angled triangle in quadrants one, label the angle at the origin as x
Reflect the triangle in the origin, it will fall into the third quadrant.
notice both the run and rise of the first have now become negative, while the hypotenuse is always considered positive.
so in 1st: sinx = opposite/hypotenuse
in third quadrant:
sinx = -opposite/hyptenuse = -sinx

test with some value of x
e.g. let x = 23.89°
sin 23.89° = .404982013..
sin(23.89+180)
= sin 203.89° = -.404982013

Answer 2

Thank you Reiny

Answer 3

sin(x+pi) = sinx cos pi + cosx sin pi = sinx(-1)+cosx(0) = -sinx