7.4 Sum-to-Product and Product-to-Sum Formulas

Trigonometric formulas are powerful tools for simplifying complex expressions. Sum-to-product and product-to-sum formulas let you switch between multiplying trig functions and adding them, making calculations easier.

These formulas are super useful in math and physics. They help solve tricky equations, find exact values, and model real-world situations. Mastering them opens up a whole new world of problem-solving possibilities.

Sum-to-Product and Product-to-Sum Formulas

Products to sums conversion

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• Convert products of trigonometric functions into sums or differences using product-to-sum formulas
  • Derived using the sum and difference formulas for sine and cosine
  • Identify the product of trigonometric functions ($\sin A \cos B$, $\cos A \sin B$, $\cos A \cos B$, or $\sin A \sin B$)
  • Substitute the appropriate values for A and B into the corresponding formula:
    • $\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$
    • $\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]$
    • $\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$
    • $\sin A \sin B = -\frac{1}{2}[\cos(A+B) - \cos(A-B)]$
  • Simplify the resulting expression using trigonometric identities and algebraic techniques

Sums to products transformation

• Transform sums or differences of trigonometric functions into products using sum-to-product formulas
  • Derived using the sum and difference formulas for sine and cosine
  • Identify the sum or difference of trigonometric functions ($\sin A + \sin B$, $\sin A - \sin B$, $\cos A + \cos B$, or $\cos A - \cos B$)
  • Substitute the appropriate values for A and B into the corresponding formula:
    • $\sin A + \sin B = 2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})$
    • $\sin A - \sin B = 2 \cos(\frac{A+B}{2}) \sin(\frac{A-B}{2})$
    • $\cos A + \cos B = 2 \cos(\frac{A+B}{2}) \cos(\frac{A-B}{2})$
    • $\cos A - \cos B = -2 \sin(\frac{A+B}{2}) \sin(\frac{A-B}{2})$
  • Simplify the resulting expression using trigonometric identities and algebraic techniques

Applications of trigonometric formulas

• Simplify complex trigonometric expressions using sum-to-product and product-to-sum formulas
  • Identify the products, sums, or differences of trigonometric functions within the expression
  • Apply the appropriate formula to convert the products into sums or differences, or vice versa
  • Combine with other trigonometric identities (such as $\sin^2 x + \cos^2 x = 1$, double angle formulas, etc.) to further simplify
• Evaluate trigonometric expressions for specific angle values
  1. Substitute the given angle values into the expression
  2. Apply the appropriate sum-to-product or product-to-sum formula to simplify the expression
  3. Evaluate the resulting expression using a calculator or by hand, if possible
• Solve trigonometric equations by applying sum-to-product and product-to-sum formulas
  • Identify the equation to be solved ($\sin x + \sin 2x = 1$, $\cos x \cos 2x = \frac{1}{2}$, etc.)
  • Apply the appropriate formula to convert the products into sums or differences, or vice versa
  • Solve the resulting equation using algebraic techniques and trigonometric identities

Advanced Trigonometric Concepts

• Understand the relationship between trigonometric functions and periodic functions
• Use radian measure to express angles in trigonometric formulas and equations
• Apply Euler's formula to connect complex exponentials with trigonometric functions
• Utilize trigonometric identities to simplify expressions and solve equations involving sum-to-product and product-to-sum formulas