# How can I recall reciprocal trig functions on the unit circle very quickly?

I'm self-studying pre-calculus right now and have run up on a huge bump in the hill. I'm really slow at recalling the values of trig functions on the unit circle given an angle. For instance csc(π/3) may take me upwards of 20-30 seconds because I'm so slow at recalling it. I'm usually very quick when it comes to math, but for the past two days I've tried improving my speed and have only made slight increases in speed.

My current algorithm goes like this :
1.) Recall the non-reciprocal function's ratio
2.) Flip this
3.) Rationalize the ratio if needed

My problem is for some reason panicking and suddenly taking 5 seconds to figure out if csc is the reciprocal of sin or cos, even though I have practiced forever that it is sin's. Then sometimes, despite me knowing the sin and cos ratios very well, struggle with getting my brain to remember where I'm searching (like what angle I'm looking for). Finally, I sometimes freeze up and can't do the simple rationalization once I flip it.

I made flash cards but they don't seem to help me so much. I'm really frustrated and stuck and seem to have some mental problem happening. Usually this stuff is really easy for me to get good at, but I'm having an incredibly difficult time getting good at it. Do you have any advice? Thanks

• Two days is not long enough to build speed. You seem do be progressing well at this point. Just keep on practicing with exercises over time, and you'll get it. I don't mean to diminish what seems to be a frustrating experience for you. But don't expect too much from yourself too soon. – Namaste Aug 1 '17 at 19:33

I got a Ph.D. in math in 1982, and have been a mathematician and/or computer scientist ever since. I, too, cannot tell you $\csc(\pi/3)$ very fast -- probably takes me 3 or 4 seconds each time I need it.

The good news? I've probably needed it (or things like it) about 20 times in my 35-year (so far) career. So even if it took me 30 seconds, that'd only be ten minutes wasted.

Short answer: You're asking the wrong question. It should be "Do I need to be able to compute these things quickly?" and the answer is "No, unless you're messing with something where they come up a lot, and in that case, you'll get quicker at them."

• Thanks for replying. My future calculus teacher wants us to get really quick at recalling the ratios (I remember walking in on his calculus class and seeing they would have a minute quiz or two where they had to solve 6 or so) so I want to be able to comply to his standards. I guess they would come up a lot in that case? The problem is I've been messing with these functions for a few weeks now and I'm still super slow at them, and I need to force myself to get quicker since everyone else in my future class is probably 10x faster – Dinoswarleafs Aug 1 '17 at 19:25
• Hunh. I suppose you don't have a choice among multiple teachers, do you? Because if this is what your teacher thinks is important, I'm not sure that you've got a very wise mentor on your hands. Still, knowing the sines and cosines of the four cardinal directions and the sine and cosine of $\pi/4, \pi/3, \pi/6$ is enough to get you where you need to go -- just remember that secant goes with COsine and COsecant goes with sine because...the folks who named them were trying to annoy you. :) – John Hughes Aug 1 '17 at 20:26
• I use this. There must be one c and one s. The reciprocal of sin is csc and the reciprocal of cos is sec. – steven gregory May 7 '18 at 1:40

I want to elaborate on John Hughes' answer. Background: a graduate math student, planning on proceeding to a Ph.D. in industrial engineering.

Some basic trigonometric ratios with secants and tangents take me a while to recall quickly. Of course, every student is different and there is no surefire way to increase speed. The best advice I can give is practice.

If you plan on taking AP Calculus, note that the multiple choice sections are not terribly long, and any calculation, especially with trigonometry, will be "nice" angles. The free-response section is the same way. (I took the AP exam in 2012.)

The point I make with my students is not speed, but understanding. So long as they understand the process, I'm satisfied. This is not AGDQ - you don't have to save the frames every time.

Practice, practice, practice.

I recommend drawing a unit circle, maybe using a protractor to get accurate angles, and draw in the sixteen main points: four compass angles, four odd multiples of pi/4, eight multiples of pi/6. Label each of sixteen points with the angle, the cosine and sine, perhaps the tangent. The other three types are reciprocals of these. Probably worth writing in some of the negative angles for the fourth quadrant. As you seem to indicate that it is the reciprocals that cause trouble, write those in. The point here, the only point, is you writing it.

Then do it again.

And again.

And again tomorrow.

The importance of this is your hands doing the drawing.

This is, of course, what i did, call it 45 years ago. It worked.