[Math] How to determine the exact value of $\sin(585^\circ)$
trigonometry
I'm clueless on this question. Could someone explain how to do it?
Best Answer
When calculating in degrees, $\sin$ is periodic with a period of 360 degrees. Hence
$$\sin(585^\circ)=\sin(225^\circ).$$
In particular, $\sin(x+180^\circ)=-\sin(x)$.
Hence $$\sin(225^\circ)=\sin(45^\circ+180^\circ)=-\sin(45^\circ).$$
On the other hand, we know that $\sin(45^\circ)=\cos(45^\circ)=\frac{1}{\sqrt{2}}$.
Hence
$$\sin(585^\circ)=-\frac{1}{\sqrt{2}}.$$
Best Answer
When calculating in degrees, $\sin$ is periodic with a period of 360 degrees. Hence $$\sin(585^\circ)=\sin(225^\circ).$$ In particular, $\sin(x+180^\circ)=-\sin(x)$. Hence $$\sin(225^\circ)=\sin(45^\circ+180^\circ)=-\sin(45^\circ).$$ On the other hand, we know that $\sin(45^\circ)=\cos(45^\circ)=\frac{1}{\sqrt{2}}$. Hence $$\sin(585^\circ)=-\frac{1}{\sqrt{2}}.$$