91Ó°ÊÓ

The given trigonometric function$\sin 45°+\sin 135°+\sin 225°+\sin 315°$

Recall that the unit circle gives $(x,y)=(\cos \mathrm{Î\pm },\sin \mathrm{Î\pm })$ for all the common angles:

Consider$\sin 45°$

Let$\mathrm{Î\pm }=45°$

From the above unit circle,

The angle$\mathrm{Î\pm }=45°$ intersects the unit circle at$\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

The pair of coordinates corresponding to $45°$is$\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

Since the second coordinate gives$\sin 45°$

Therefore the value of$\sin 45°=\frac{\sqrt{2}}{2}$ is$\frac{\sqrt{2}}{2}$

Consider$\sin 135°$

Let$\mathrm{Î\pm }=135°$

From the above unit circle,

The angle$\mathrm{Î\pm }=135°$ intersects the unit circle at$\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

The pair of coordinates corresponding to $135°$is$\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

Since the second coordinate gives$\sin 135°$

Therefore the value of $\sin 135°$is$\frac{\sqrt{2}}{2}$

Next,consider$\sin 225°$

Let$\mathrm{Î\pm }=225°$

From the above unit circle,

The anglerole="math" localid="1647676365927" $\mathrm{Î\pm }=225°$ intersects the unit circle at$\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$

The pair of coordinates corresponding to $225°$is$\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$

Since the second coordinate gives$\sin 225°$

Therefore the value of $\sin 225°$is$-\frac{\sqrt{2}}{2}$

Consider$\sin 315°$

Let$\mathrm{Î\pm }=315°$

From the above unit circle,

The angle $\mathrm{Î\pm }=315°$intersects the unit circle at$\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$

The pair of coordinates corresponding to $315°$is$\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$

Since the second coordinate gives$\sin 315°$

Therefore the value of $\sin 315°$is$-\frac{\sqrt{2}}{2}$

Finally add all these four exact values to obtain the required solution

Therefore,

$$\begin{gathered} \sin 45°+\sin 135°+\sin 225°+\sin 315°=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}+\left(-\frac{\sqrt{2}}{2}\right)+\left(-\frac{\sqrt{2}}{2}\right) \\ =0 \end{gathered}$$

Hence answer is$0$