The R Addition Formulas

The R Addition Formulas

Definition

• R-Formulas or Product-to-Sum Formulas can be a useful tool in simplifying some trigonometric expressions.
• They are derived from the sum and difference identities and can express a product of functions as an addition or subtraction.

Key Formulas

• Sin(u) · Sin(v) = ½ [Cos(u-v) - Cos(u+v)]
• Cos(u) · Cos(v) = ½ [Cos(u-v) + Cos(u+v)]
• Sin(u) · Cos(v) = ½ [Sin(u+v) + Sin(u-v)]

Working with R-Formulas

• One common use of r-formulas is to help integrate products of sines and cosines.
• When given a problem involving products of trig functions, look for opportunities to use the r-formulas.
• Also remember to factor out constants before applying r-formulas, this will make your work easier.

Limitations

• R-formulas can sometime make calculations more complex. In some cases, other manipulations like trig identities might simplify the problem more effectively.

Practice Problems

• Rewrite the expression 2Sin(3x)Cos(4x) using the r-formula.
• Simplify the integral of Sin(x)Cos(x) dx using the r-formula.
• Find the integral of 3Sin(2x)Cos(2x) dx by applying the r-formula.