TRIG_TR3

This transformation applies induced trigonometric identities that normalize signs and argument structure, especially for negative arguments and periodic shifts.

Typical rules include odd/even behavior:

\sin(-x)=-\sin(x),\quad \cos(-x)=\cos(x)

These rewrites can expose additional simplification opportunities.

Excel Usage

=TRIG_TR3(expression)

expression (str, required): Trigonometric expression for sign and induced-identity normalization (expression string).

Returns (str): Transformed expression as a string, or an error message.

Example 1: Normalizes odd sine sign

Inputs:

expression
sin(-x)

Excel formula:

=TRIG_TR3("sin(-x)")

Expected output:

"-sin(x)"

Example 2: Normalizes even cosine sign

Inputs:

expression
cos(-x)

Excel formula:

=TRIG_TR3("cos(-x)")

Expected output:

"cos(x)"

Example 3: Simplifies induced shift expression

Inputs:

expression
cos(y - x*(y - x))

Excel formula:

=TRIG_TR3("cos(y - x*(y - x))")

Expected output:

"cos(x*(x - y) + y)"

Example 4: Normalizes mixed negative trig arguments

Inputs:

expression
sin(-a) + cos(-b)

Excel formula:

=TRIG_TR3("sin(-a) + cos(-b)")

Expected output:

"-sin(a) + cos(b)"

Python Code

from sympy import sympify
from sympy.simplify.fu import TR3 as sympy_tr3

def trig_tr3(expression):
    """
    Normalize trig signs and induced odd/even identity forms.

    See: https://docs.sympy.org/latest/modules/simplify/fu.html#sympy.simplify.fu.TR3

    This example function is provided as-is without any representation of accuracy.

    Args:
        expression (str): Trigonometric expression for sign and induced-identity normalization (expression string).

    Returns:
        str: Transformed expression as a string, or an error message.
    """
    try:
        expr_text = expression.replace("^", "**") if isinstance(expression, str) else expression
        expr_obj = sympify(expr_text)
        result = sympy_tr3(expr_obj)
        return str(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

expression *

Trigonometric expression for sign and induced-identity normalization (expression string).