Question on trigonometric notation (specifically wrt parentheses)

Question

1. $\sin x = \sin(x)$
2. $\sin xy = \sin(x*y)$
3. $\sin xy = y\sin(x)$
4. $\sin x*y = \sin(x*y)$
5. $\sin x*y = y\sin(x)$

#1 is a given, since the whole field discusses "trigonometric functions"

If #2 were the conventional understanding of the notation, which seems to be the case from a cursory look at various textbooks, then #3 would be at odds, from a semantic perspective, with #2, in which case it would be odd for anyone to assume that #3 should be the correct way to understand the given notation. Am I correct in assuming that #2, not #3, is the norm and nobody really serious about the topic in question assumes #3?

If #2 were a variant of #4, then there would be no confusion as to the meaning of the notation; however I've yet to see any expression that would utilize the * as in #4 to express sin(xy). Am I correct to assume that #4 is not part of conventional notation?

I apologize if this seems like a strange question.

Answer 1

Let's analyze the expressions:

1. $\sin(x) = \sin(x)$ - This is a trigonometric identity and is correct.

2. $\sin(xy) = \sin(x \cdot y)$ - This is the conventional understanding where xy represents the product of x and y.

3. $\sin(xy) = y \sin(x)$ -this seems to be an alternative interpretation wTere xy is understood as x times y. This is not a conventional interpretation and could lead to confusion.

4. $\sin(x \cdot y) = y \sin(x)$ -This uses the multiplication symbol (⋅) explicitly and is a valid expression, but it's not a standard notation in trigonometry.

In standard mathematical notation, when variables are written next to each other, it generally implies multiplication. The second expression (#2) is the conventional and widely accepted interpretation, where xy is understood as the product of x and y. The third expression (#3) deviates from the convention and may lead to confusion.

The use of the asterisk (*) for multiplication is more common in computer programming and some specific mathematical contexts, but it's not the standard notation in general mathematics. So, #4 is not a conventional notation either.