From trigonometry we know that:
if [tex] sin(\theta)=sin(\alpha) [/tex]
then, [tex] \theta=n\pi+(-1)^n\alpha [/tex] (where [tex] n [/tex] is an integer)
This can be rewritten in degrees as:
[tex] \theta=n(180^{\circ})+(-1)^n\alpha [/tex].............(Equation 1)
Now, in our case, [tex] \alpha=-230^{\circ} [/tex]
Therefore, (Equation 1) can be written as:
[tex] \theta=n(180^{\circ})+(-1)^n(-230^{\circ}) [/tex]..........(Equation 2)
Now, to find the correct options all that we have to do is replace n by relevant integers and find the values of [tex] \theta [/tex] that match.
For n=2, (Equation 2) gives us: [tex] \theta=2\times 180^{\circ}+(-1)^2(-230^{\circ})=360^{\circ}-230^{\circ}=130^{\circ} [/tex].
Thus, [tex] sin(230^{\circ})=sin(130^{\circ}) [/tex]
Now, we know that: [tex] -sin(-50^{\circ})=sin(50^{\circ}) [/tex]
Let n=-1, then:
[tex] \theta=(-1)\times 180^{\circ}+(-1)^{-1}(-230^{\circ})=-180^{\circ}+230^{\circ}=50^{\circ} [/tex]
Thus, [tex] sin(-230^{\circ})=-sin(-50^{\circ}) [/tex]
Likewise, [tex] sin(-230^{\circ})=sin(50^{\circ}) [/tex]
Only the last option [tex] sin(-50^{\circ}) [/tex] will never match [tex] sin(-230^{\circ}) [/tex] because no integral value of [tex] n [/tex] will ever give [tex] \theta=-50^{\circ} [/tex]
Thus the last option is the correct option.
