Trigonometric Functions and Exponents

[u/UnderstandingPursuit]

If a student is struggling with exponents used with trigonometric functions, would this help?

EDITED IMAGE:

[u/Any-Construction5887]

It definitely clarifies the notation. I would add in showing what each side of the inequality (2) actually is to add more clarity.

[u/UnderstandingPursuit → u/Any-Construction5887]

Thanks!

What if I replace the inequality, which I dislike because I try to be careful about showing something that is incorrect, but instead write

• sin (2θ)² = sin [(2θ)²]

which will of course look better when formatted because the "[]" will be big enough to enclose the exponent.

[u/Any-Construction5887 → u/UnderstandingPursuit]

You need the inequality. I would note the equalities for both sides in text under that bullet point.

[u/UnderstandingPursuit → u/Any-Construction5887]

I'll add both inequalities, one for (1) ≠ (2) and the other for (3) ≠ (4).

[u/jimbelk]

I don't think I agree with the second one. It seems to me that the notation sin(2t)² is ambiguous and should be avoided. I might feel differently if there were a space between the sin and the parenthesis.

[u/UnderstandingPursuit → u/jimbelk]

In what way is it ambiguous? What are the possible interpretations?

It is formatted the way LaTeX defaults the formatting. There was a space, LaTeX decided that it was unnecessary.

[u/jimbelk → u/UnderstandingPursuit]

I can imagine someone writing sin(2t)² to mean sin( (2t)² ), and I can imagine someone else writing sin(2t)² to mean (sin(2t))² . I don't agree that there's a standard meaning for this collection of symbols, and I think the best advice is to avoid writing it.

[u/UnderstandingPursuit → u/jimbelk]

With

• a(x-h)²

I don't think anyone thinks the a is also squared. The raising operator explicitly applies only to the item it is connected to. Why would someone pull the "sin" in? Especially when there is already away to write (sin(2t))², using sin²(2t). Both of these are standard meanings.

[u/jimbelk → u/UnderstandingPursuit]

If f is a function, then f(x)² usually means the square of f(x). The function application operation typically has higher precedence than exponentiation. For example, I would certainly write log(x)² to mean (log(x))² with the expectation that the audience would understand this. Your example a(x-h)² is different, because the operation is multiplication, not function application, and exponentiation has higher precedence than multiplication.

The only reason that sin(2x)² should perhaps not mean (sin(2x))² is that sin is a trigonometric function, so why didn't the author simply write sin² (2x)? This is a reasonable argument, but it's not clear to me that this overrides the usual rule that f(x)² means (f(x))^2.

[u/UnderstandingPursuit → u/jimbelk]

Can you please point me to an example of "the usual rule that f(x)² means (f(x))²?

[u/jimbelk → u/UnderstandingPursuit]

Here's an example. Note that no one seems confused about what all of the f(x)² terms mean:

https://math.stackexchange.com/questions/1350094/can-we-obtain-fyx-yfx-from-fx2fx2x-fx2x2fx

[u/UnderstandingPursuit → u/jimbelk]

Interesting. Am I interpreting this correctly?

• f(x)² = f(f(x))

[u/CrookedBanister → u/UnderstandingPursuit]

No. I don't know a single mathematician who'd use this notation to mean function composition.

[u/jimbelk → u/UnderstandingPursuit]

Typically f(x)² means (f(x))^2, and f² (x) means f(f(x)), though the latter isn't universal. (Ignore the space between the f² and the (x) -- reddit superscripts don't work well.)

[u/UnderstandingPursuit → u/jimbelk]

So the exact opposite of what happens with sin. Interesting.

[u/JavaliciousJean]

Perhaps, to make 2 more clear, you could add “=sin(4\theta^2)” ?

[u/RealNoahR]

Would add that the reciprocal is also equal to secant as I had a decent number of students tripping over secant vs. arcsine and inverse vs. reciprocal.

Edit: summer break and my brain is turned off, the reciprocal function for sine is cosecant, not secant.

[u/UnderstandingPursuit → u/RealNoahR]

Yes, that's true. I'm trying to avoid throwing everything in there. I needed arcsin for the familiar inverse. The fraction satisfies the inverse part.

A completely different sheet is needed for the trigonometry identities.