7.2 Sum and Difference Identities

Trigonometric sum and difference identities are powerful tools for simplifying complex expressions and solving equations. These formulas allow us to break down compound angles into simpler components, making calculations easier and more manageable.

By mastering these identities, we gain a deeper understanding of how trigonometric functions behave when angles are added or subtracted. This knowledge is crucial for tackling advanced problems in trigonometry and preparing for more advanced mathematical concepts.

Deriving and Applying Sum and Difference Identities

Sum and difference formulas

• Geometric approach employs unit circle and angle addition properties visualizing angle addition on coordinate plane
• Cosine sum formula $\cos(A + B) = \cos A \cos B - \sin A \sin B$ leads to cosine difference formula
• Sine sum formula $\sin(A + B) = \sin A \cos B + \cos A \sin B$ generates sine difference formula
• Tangent sum formula $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ derived using quotient identity and previous formulas
• Tangent difference formula follows similar derivation process

Application of trigonometric identities

• Recognize expressions $\sin(A ± B)$, $\cos(A ± B)$, or $\tan(A ± B)$ for identity application
• Cosine formulas: $\cos(A + B) = \cos A \cos B - \sin A \sin B$, $\cos(A - B) = \cos A \cos B + \sin A \sin B$
• Sine formulas: $\sin(A + B) = \sin A \cos B + \cos A \sin B$, $\sin(A - B) = \sin A \cos B - \cos A \sin B$
• Tangent formulas: $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$, $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
• Evaluate expressions with special angles (30°, 45°, 60°) or use calculator for non-standard values

Simplification with identities

• Break down compound angles into sums or differences
• Combine like terms after applying identities
• Factor common terms and rationalize denominators when necessary
• Verify simplifications using other trigonometric identities
• Apply sum and difference identities alongside other trig identities for comprehensive simplification

Solving equations with identities

• Rewrite equations using appropriate sum or difference identity
• Isolate variable through algebraic techniques (combining terms, factoring)
• Solve using inverse trigonometric functions
• Determine general solution and restrict to specific interval if needed
• Verify solutions by substitution into original equation
• Tackle equations with multiple angles by rewriting in terms of a single angle using sum and difference identities