Question

Use a sum or difference identity to find the exact value of :

Answer 1

`\begin{gathered} \sin 285\text{ } \\ 285\text{ can be split into 225 and 60} \end{gathered}`
`\sin (225+60)`

Using the rule

`\sin (x+y)=\sin x\cos y+\cos x\sin y`
`\begin{gathered} \sin (225+60)=\sin 225\cos 60+\cos 225\sin 60 \\ \end{gathered}`

Sine is negative in the third quadrant therefore,

`\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=}(1)/(2) \\ -\frac{\sqrt[]{2}}{2}((1)/(2))+\cos 225\sin 60 \end{gathered}`

Let us find the other side

`\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}`

Bring everything together

`\begin{gathered} -\frac{\sqrt[]{2}}{2}((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}`