It is convenient, as adults, to use the notation “ ” to refer to the value of the square of the sine function. However, rushing too fast to this abbreviated notation for trigonometric functions leads to incorrect understandings of how functions are manipulated, which can lead students to think that is short for “ ” and to incorrectly divide out the variable, “ .”
To reduce these types of common notation-driven errors later, this curriculum is very deliberate about how and when we use abbreviated function notation for sine, cosine, and tangent:
- In geometry, sine, cosine, and tangent are thought of as the value of ratios of triangles, not as functions. No attempt is made to describe the trigonometric ratios as functions of the real number line. Therefore, the notation is just an abbreviation for the “sine of an angle” ( ) or “sine of an angle measure” ( ). Parentheses are used more for grouping and clarity reasons than as symbols used to represent a function.
- In Algebra II, to distinguish between the ratio version of sine in geometry, all sine functions are notated as functions: is the value of the sine function for the real number , just like is the value of the function for the real number . In this grade, we maintain function notation integrity and strictly maintain parentheses as part of function notation, writing, for example, , instead of .
- By pre-calculus, students have had two full years of working with sine, cosine, and tangent as both ratios and functions. It is finally in this year that we begin to blur the distinction between ratio and function notations and write, for example, as the value of the square of the sine function for the real number , which is how most calculus textbooks notate these functions.