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I have been working with students with trig identities, but cannot find good resources for problems about trigonometric identities and proofs. Often any resources I do find are short and one note - each problem set only focuses on one identity.

A bit of a shame as my old trig book had quite a few open-ended challenging problems in addition to one-note problems for getting familiar with identities. I can no longer remember this book, however.

I'd appreciate if anyone has any resources to recommend with a wider variety and challenge of problems! Preferably online to be accessible, but good books work too.

I also am not opposed to creating my own problems - but have never created problem sets before. Any tricks for that, or is it as simply as piling a few select identities on one another and shuffle around terms to create tricky problems?

Thanks for any answers!

PinkyWay
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Andrew
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I cannot say for the books (partially because it has been answered and partially because I don't know any). However, I can help you in terms of creating your own problems. As a person who used to do this on a very regular basis, here are some tips:

  1. First, you need to figure out to what audience are you setting your problems and how frequently you plan to create trigonometric problems.
  2. Then, you must figure out how hard must the problems must be (for example, the first set can be easy, the second set medium, the third set hard, the fourth set extremely difficult and the fifth set deceptively impossible (I specialise in 3-5 but you can do whatever you feel makes you comfortable))
  3. Write down whatever you are thinking of without hesitation! This is crucial (for example, I was thinking a few days ago: $$\fbox{($x_1x_2x_3)^n=k^n-k^{n-a}, (x_1x_2+x_1x_3+x_2x_3)^n=k^{n-a}+k^{2n}, (x_1+x_2+x_3)^n=$?}$$ and I just instantly wrote it down without hesitating because hesitation often leads to this process getting interrupted a lot, which won't help in the long-term - I know from experience.)
  4. You must then solve it yourself. At this point, the problem you wrote down may look daunting, but trust me, by solving it yourself, it will eventually look like nothing once you apply your knowledge and see how far you get. (You can ask for help but only after you have at least gotten somewhere yourself.)
  5. Tweak it if necessary. If, for some reason, the problem doesn't give you the intended answer, go through all stages of working (must be done on pen and paper or a Word document first) and double-check everything (with a friend, WolframAlpha, Mathematica, etc. - whatever you feel will help.). If something is wrong, trace it back and figure out whether it was human error (e.g. you put the wrong number or expression down) or if it was an inherent error (as in, it was always going to lead to this because it was in the original problem statement itself). If it was a human error, rectify it and solve it again - you should get the desired answer. If it was an inherent error, change the statement and repeat tip 4 to see where that leads. Then check for human and inherent errors. If it was a human error, fix it and try again. If it was an inherent error, keep repeating the last 3-4 lines of this tip until you get it right with no errors.
  6. If you are making a problem set or loads of problem sets, repeat the above 5 tips until you have completed the problem set(s).

Anything in bold is a really crucial tip and one that should always stay in your mind.

Good luck making problems!

Yajat Shamji
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