De Moivre's Therom - Roots of Complex Numbers- reference angle

Question

Regarding De Moivre's Therom Roots of Complex Numbers,

1. during calculation, how can one quickly know that $$ cos225 = -\frac{\sqrt{2}}{2}$$ $$ sin225 = -\frac{\sqrt{2}}{2}$$

2. can we also use calculator?

i read this, but it seems time consuming.

https://socratic.org/questions/how-do-i-find-the-value-of-cos-225#168399

Thanks for the help!

Answer 1

An isosceles right-angled triangle gives $\sin45^\circ=\cos45^\circ=1/\sqrt{2}=\sqrt{2}/2$. Both functions multiply by $-1$ if you add $180^\circ$ to their argument, as it's half a period.