TRIG_EXPAND
This function expands trigonometric expressions by applying angle-sum and related expansion identities. It is useful for converting compact forms into additive and multiplicative components.
For example, one key identity is:
\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)
The deep option controls whether expansion recursively traverses nested subexpressions.
Excel Usage
=TRIG_EXPAND(expression, deep)expression(str, required): Trigonometric expression to expand (expression string).deep(bool, optional, default: true): Apply expansion recursively to nested terms (true/false).
Returns (str): Expanded expression as a string, or an error message.
Example 1: Expands sine of a sum
Inputs:
| expression |
|---|
| sin(a + b) |
Excel formula:
=TRIG_EXPAND("sin(a + b)")Expected output:
"sin(a)*cos(b) + sin(b)*cos(a)"
Example 2: Expands cosine of a sum
Inputs:
| expression |
|---|
| cos(a + b) |
Excel formula:
=TRIG_EXPAND("cos(a + b)")Expected output:
"-sin(a)*sin(b) + cos(a)*cos(b)"
Example 3: Expands trig factor inside a larger product
Inputs:
| expression |
|---|
| sin(x + y)*(x + y) |
Excel formula:
=TRIG_EXPAND("sin(x + y)*(x + y)")Expected output:
"(x + y)*(sin(x)*cos(y) + sin(y)*cos(x))"
Example 4: Disables deep recursion when requested
Inputs:
| expression | deep |
|---|---|
| sin((a + b) + c) | false |
Excel formula:
=TRIG_EXPAND("sin((a + b) + c)", FALSE)Expected output:
"(-sin(b)*sin(c) + cos(b)*cos(c))*sin(a) + (sin(b)*cos(c) + sin(c)*cos(b))*cos(a)"
Python Code
from sympy import sympify, expand_trig as sympy_expand_trig
def trig_expand(expression, deep=True):
"""
Expand trigonometric compound-angle expressions into component terms.
See: https://docs.sympy.org/latest/modules/core.html
This example function is provided as-is without any representation of accuracy.
Args:
expression (str): Trigonometric expression to expand (expression string).
deep (bool, optional): Apply expansion recursively to nested terms (true/false). Default is True.
Returns:
str: Expanded 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_expand_trig(expr_obj, deep=deep)
return str(result)
except Exception as e:
return f"Error: {str(e)}"