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[MagmaMcFry]

Trigonometric proofs? Do you mean proofs of statements that use trigonometric functions, or geometric proofs using trigonometry?

[BlindKitty]

@Ligthingfalcon: Or do you mean trigonometric identities? I remember that name from high school, and there were more than on this wikipedia article.

[Lightningfalcon]

Proofs that require identities to solve.
Things like cos^2=1-sin^2, but a bit more complicated.
Also problems such as cos(x)=tan(x)+sin(x)^2, where you have to solve for x.   

[MagmaMcFry]

You could take the Wikipedia page on trigonometric identities and try and prove all the stated identities by using other identities.

[Helgoland]

How about Taylor expansions?

[Skyrunner]

What is the graph of

y=1/5*(x-4)^(5/3)+2 (x-4)^(2/3)
?
Wolfram Alpha has it to be complex when x <4, but my book and my ti84 has it real for all x.

[Urist Imiknorris]

I'm betting WA interpreted it as 1/(5*(x-4)^(5/3)). It's real for all x.

[Skyrunner]

I'll have your bet. What did you bet?

If you pasted it into WA, you can see it interpreted it the way it should have been interpreted, with 1/5 being multiplied by (x-4)^(5/3).

[Tylui]

ptw!

[Helgoland]

What is the graph of

y=1/5*(x-4)^(5/3)+2 (x-4)^(2/3)
?
Wolfram Alpha has it to be complex when x <4, but my book and my ti84 has it real for all x.

For me, it interpreted it as (x-4) O.o
Just whip out some paper and start plotting - hint: It's the graph of (1/5)*x^(5/3) + 2*x^(2/3) , just moved four units to the right.

[Skyrunner]

Copy paste the entire thing into WA. It says that the plot is complex for x <4.

[da_nang]

IIRC, there's a minor detail about x^p for x<0 when p is not an integer.

[MagmaMcFry]

Skyrunner, when you input your formula, WA says "Assuming principal root, use real-valued root instead" at the top of the results box. Click the hypertext and you should get correct results instead.

[Skyrunner]

Accursed fine text!

So what's a principal root? I noticed that if you take the number Wolfram says is (-4)^(3/2) (something something i) and square it then sqrt it, it's equal to the real-value root times minus one!

[Helgoland]

AFAIK, principal roots are roots in the complex plane - there's usually many of them,and they're all on a circle.
For example, the principal third roots of 2 would be 2^(1/3), 2^(1/3)*e^(i*120°), and 2^(1/3)*e^(i*240°).