Question About Unit Circle (CircularFunction) of a Trig Func

[basty]

Please take a look below example (the attached image below).

How do I know that the angle ##\sin (\frac{7π}{4})## is corresponds to the coordinates ##(\frac{\sqrt {2}}{2}, -\frac{\sqrt{2}}{2})##?

I know that ##\frac{7π}{4}## is 315°.

 

[BvU]

Did you draw a unit circle and mark e.g. the ##7\pi \over 4## angle ?

 

[basty]

Did you draw a unit circle and mark e.g. the ##7\pi \over 4## angle ?

Should I?

 

[BvU]

Yes

 

[Ssnow]

subtract ## 2\pi## that is ##\frac{7}{4}\pi -2\pi=\frac{7-8}{4}\pi=-\frac{\pi}{4}##...

 

[HallsofIvy]

If you draw a line from (0, 0) with length 1 and making angle [itex]\theta[/itex] with the x-angle and drop a perpendicular to the x-axis, then the distance to the foot of that perpendicular, along the x-axis is the "near side" of a right triangle with angle [itex]\theta[/itex] and hypotenuse 1. Similarly, the length of the perpendicular, parallel to the y-axis, is the "opposite side".

 

[aikismos]

Well, I think any high school teacher I knew when I was teaching would have a simple answer:

Memorize the unit circle (which isn't so hard to do if notice the angular symmetries and remember the mnemonic device All Students Take Calculus in order to remember the signs).