[tex]\begin{gathered} \sin 285\text{ } \\ 285\text{ can be split into 225 and 60} \end{gathered}[/tex][tex]\sin (225+60)[/tex]
Using the rule
[tex]\sin (x+y)=\sin x\cos y+\cos x\sin y[/tex][tex]\begin{gathered} \sin (225+60)=\sin 225\cos 60+\cos 225\sin 60 \\ \end{gathered}[/tex]
Sine is negative in the third quadrant therefore,
[tex]\begin{gathered} -(\sin 45)\cos 60+\cos 225\sin 60 \\ \sin \text{ 45=}\frac{\sqrt[]{2}}{2}\text{ then the negative sign} \\ -\frac{\sqrt[]{2}}{2} \\ -\frac{\sqrt[]{2}}{2}\cos 60+\cos 225\sin 60 \\ \cos \text{ 60=}\frac{1}{2} \\ -\frac{\sqrt[]{2}}{2}(\frac{1}{2})+\cos 225\sin 60 \end{gathered}[/tex]
Let us find the other side
[tex]\begin{gathered} \cos \text{ 45=}\frac{\sqrt[]{2}}{2} \\ cos\text{ is negative in the third quadrant } \\ -\frac{\sqrt[]{2}}{2} \\ \sin \text{ 60=}\frac{\sqrt[]{3}}{2} \\ \end{gathered}[/tex]
Bring everything together
[tex]\begin{gathered} -\frac{\sqrt[]{2}}{2}(\frac{1}{2})-\frac{\sqrt[]{2}}{2}(\frac{\sqrt[]{3}}{2}) \\ -\frac{\sqrt[]{2}}{4}-\frac{\sqrt[]{6}}{4}=\frac{-\sqrt[]{2}-\sqrt[]{6}}{4}=-0.965925826\ldots... \end{gathered}[/tex]
