The unit circle and circular functions

I am attempting to do my homework but my book got lost in the mail, I have a test on Monday, and I only have the homework problems and my meticulous notes from class.

The next set of homework asks for the exact circular function value for $\sin (7\pi/6)$.

How do I figure this out without a calculator or diagram?

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Do you know how to calculate $\sin(\pi+\theta)?$ –  Jack Jun 11 '11 at 22:30
No, I do not know what that means. –  Adam Jun 11 '11 at 22:31
@Adam: you may want to read this. –  Jack Jun 11 '11 at 22:34
I was just going through that, does that mean I have to memorize them all? –  Adam Jun 11 '11 at 22:36
@Jack: Adam has been referred to Wikipedia's identities a number of times now. @Adam: I thought you just had a test? (math.stackexchange.com/questions/44653/tan-sec-test-questions)... Adam: You seem to be asking the same sorts of questions repeatedly, showing very little learning from answers that have been provided, repeatedly, or from links you've been provided, or geometric interpretations, etc. Are you asking for more answers? –  amWhy Jun 11 '11 at 22:38

1 Answer

You don't have to memorize all of the stuff on the wikipedia page to figure out this sort of thing. For this question, you should know that $\sin(\pi/6)=1/2$ and either remember that $\sin(\pi + \theta)=-\sin(\theta)$ for any angle $\theta$, or be able to figure this out by remembering the graph of $\sin(\theta)$ and staring at the graph for a second or two. (I favour the latter approach). Or you can draw a unit circle and figure things out that way.

Anyway, taking $\theta=\pi/6$ in the formula $\sin(\pi+\theta)=-\sin(\theta)$, we get $\sin(7\pi/6)=\sin(\pi + \pi/6)=-\sin(\pi/6)=-1/2$.

If you think it's likely you'll be asked this kind of question, I would advise memorizing the values of $\sin$ and $\cos$ at $\pi/6$ and $\pi/3$ along with the graphs of $\sin$ and $\cos$, and practise using this to figure out other values of trig functions at various other angles.

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I have all the values memorized in degrees, should I bother memorizing radians or just convert them all to degrees? –  Adam Jun 11 '11 at 23:01
@Adam: good advice here. Realize that $\pi/6 = 30^\circ$. Try to develop some "fluency" with the most common angle measures: degrees <-> radians. (Simply remembering 360 degrees = $2\pi$ radians will help; from that, 180 degrees = $\pi$ radians, 90 degrees = $\pi/2$ radians, so 30 degrees is 1/3 of 90 degrees, hence 30 degrees is 1/3 of $\pi/2 = \pi/6$ radians, etc. –  amWhy Jun 11 '11 at 23:01
If it's easier for you to work with degrees, that's fine. But you'll want to be able to convert from radians to degrees, yes? And back again to radians (if that's what a test question requires). –  amWhy Jun 11 '11 at 23:04
I can convert both ways and I have a basic understanding of radians. –  Adam Jun 11 '11 at 23:19
@Adam: good! Then it's fine to work with degrees, if you're more comfortable with degrees, and values of trig functions of angles given in degrees. –  amWhy Jun 11 '11 at 23:28