Display powers of trig functions in usual way

[siwelwerd]

Closes #39

[github-actions]

🚀 Preview available 🚀

https://de2f8595.tbil.pages.dev

• source/calculus/exercises/outcomes/TI/TI3/generator.sage: https://de2f8595.tbil.pages.dev/preview/calculus/instructor/exercises/TI3/

[github-actions]

🚀 Preview available 🚀

https://eefea8c9.tbil.pages.dev

• source/calculus/exercises/outcomes/TI/TI3/template.xml: https://eefea8c9.tbil.pages.dev/preview/calculus/instructor/exercises/TI3/

[siwelwerd]

I think this is working!

However, I had to do the integration by hand and hardcode it in, so it would be helpful if someone could check the answers on these (particularly task 1). I spot checked a handful of them and they looked okay though.

[StevenClontz]

Suggested change

	also_g = k/2^(m+n)*(1+cos(2*a*x))^m*(1-cos(2*a*x))^n
	also_g = k/2^(m+n)*(1+cos(2*a*x))^m*(1-cos(2*a*x))^n
	assert f.diff() == also_g

Or something like that to ensure no errors were made? Or something that's more clearly the same as g?

[siwelwerd]

I don't think that will work, the way cosp is created sage really doesn't know that it's a power of cos. We lose a lot of that kind of thing by creating our own function.

[StevenClontz]

Hmm, it might be quicker to demo than explain what I'm trying to say, one sec

[siwelwerd]

Maybe there is a way to use assuming to do something. Maybe that's a separate issue t hough to go back and add checks like these wherever we are doing something by hand (across exercises)

[StevenClontz]

It was not quicker, so I think this is okay to use as-is.

[StevenClontz]

Does my suggestion to verify equality with code make sense? A spot-check looks good.

It's off-topic, but while we're here, should task 2 say something like "...identities to rewrite {{int}} into an integral with smaller exponents..."?

[siwelwerd]

It's off-topic, but while we're here, should task 2 say something like "...identities to rewrite {{int}} into an integral with smaller exponents..."?

I didn't love that prompt either, but was focusing on getting the powers to display properly. I'm happy for you to suggest alternative language.

[StevenClontz]

It's off-topic, but while we're here, should task 2 say something like "...identities to rewrite {{int}} into an integral with smaller exponents..."?

I didn't love that prompt either, but was focusing on getting the powers to display properly. I'm happy for you to suggest alternative language.

I'll make a quick separate PR