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I'm having trouble in Khan Academy trigonometry where there are two steps, that aren't a sequence of algebraic manipulations. I get immersed in the second step, and give that as the answer -- forgetting to apply the first step.

For example, unit circle quadrants for $\cos$ and $\sin$:

  1. the quadrant of angle $\theta$ indicates the signs of $\sin \theta$ and $\cos \theta$.

  2. given $\cos \theta$, work out $\sin \theta$, using Pythagoras.

  3. I forget to apply the correct sign.

I also keep makng silly typos, even though I know how to do it. In copying my answer from paper to the site, I omit $\pi$, I omit a negative sign. In my calculations, I use the opposite instead of the hypotenuse.

When I was in high school, I would have dismissed these errors as trivial, and the important thing was understanding. But as an adult, I feel my problem is an inability to handle complexity - even at this very mild level. It's important to be able to get the right answer; to have confidence in mathematics. I also have trouble with complexity in programming. Maybe learning a skill here for trig, will also help there...

How can I improve at handling the (mild) complexity of trig?

PS: some complicating factors: I'm tired; trying to rush through it; doing it on a small smartphone screen; KA often crashes; using google for calculations (which silently accepts mismatched parentheses).

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    $\begingroup$ Sounds like the obvious solutions would be to double-check your work, slow down, and never do coursework or anything of the like on a phone in lieu of a proper computer, to be blunt. $\endgroup$ – Eevee Trainer Apr 7 at 6:52
  • $\begingroup$ @EeveeTrainer Thanks! I do double-check my work, but yes, I could start double-checkimn the transcribing from paper to KA, too. Unfortunately, I don't have access to a PC ATM. $\endgroup$ – hyperpallium Apr 7 at 7:37
  • $\begingroup$ Watch this and learn to program: ted.com/talks/… $\endgroup$ – David G. Stork Apr 7 at 8:01
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    $\begingroup$ Get some rest and don't be in a hurry. The quickest way to take a lot more time is to do it as fast as possible. Instead of the clumsiness of TV and tiny screens, get an old fashion text book and work problems on paper. $\endgroup$ – William Elliot Apr 7 at 9:55
  • $\begingroup$ @WilliamElliot Less haste more speed, you're right. Unfortunately, I've invested many months in KA, and I feel that if I stick to KA, at least I'll have reached some standard - and crucially, without gaps. I'm trying to fill gaps from high school, but I don't know what they are... To switch now, I'd have to start at the beginning again. There's also a difficulty in evaluating paper books, when I'm not near a uni library. $\endgroup$ – hyperpallium Apr 7 at 12:24
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Techniques to avoid forgetting to apply a previous step:

  1. a checklist of the two steps, and check it at the end.

  2. represent the step algebraically, so the expression holds the "state" for you. e.g.

    • find $\cos \frac{4\pi}{3}$: After working out it's in quadrant III and therefore negative, don't try to remember that. Instead, write it in terms of quadrant I: $\cos \frac{4\pi}{3} = -\cos (\frac{4\pi}{3} - \pi)$. Now the sign is remembered for you.
  3. Khan Academy seems to do the steps in the order needed. e.g. at first, use the unit circle quadrant to determine the angle only (not the sign). Once you've worked out the absolute value of the $\sin$ or $\cos$, then work out the sign from the quadrant. By not doing the sign yet, you're not fooled into thinking you've finished with it.

The advice I took, and how it went:

slow down

I'd been aiming at 3 minutes per question, and relaxed this to 10 minutes.

  • I ended up averaging about 6 minutes. It's not that much longer (and much quicker than repeating the whole).

double-check

I made a checklist of steps (I did this after completing the problem on the website up to "check", as working out the steps needed is part of the problem). This checklist especially included the final "step" of copying the answer from paper to website. Then, I double-checked each item on the checklist.

  • tedious at first, with more detailed workings, but got easier after a few questions.

  • it caught a silly simplification error; and a silly transcription error to the website.

  • sense of confidence: no more suspense when checking an answer!

bluetooth keyboard

Using a bluetooth keyboard makes my phone a little more like a PC, for better typing, and also by increasing display area available. I made it easier to enter calculations into google, and reduced errors. (But Khan Academy insists on using its own on-screen keyboard).

tl;dr

After getting something wrong in each of my six (6) previous attempts, by slowing down and using this double-checking system, I got 100%, and felt confident of each answer.

Thanks again everyone!

UPDATE I took the unit test again (the problems change each time), and I was much faster, down to 4 minutes per question, and didn't make any mistakes for the double-checking to catch.

While this would be partly due to being fresher and having learned the material better from practice, it really felt as though the double-checking hmade my practice more effective somehow, deeper or more thorough. Perhaps because it includes an overview of the problem, a new and integative perspective.

i.e. it seems double-checking doesn't just help catch errors, but also makes my learning more effective. I had expected it to make my learning less effective, by being a crutch.

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