Sum and Difference Identities in Trigonometry: Complete Guide with Examples
Learn sum and difference identities for sine, cosine, and tangent. See examples, formulas, and step-by-step explanations for sin(A±B), cos(A±B), and tan(A±B).
Sum and difference identities are among the most powerful tools in trigonometry. They allow you to break down complex angle expressions into simpler components, evaluate trigonometric functions of non-standard angles, and derive countless other formulas. Whether you are simplifying an expression in calculus or solving a physics problem involving wave interference, mastering these identities is essential.
Key Takeaway
The six sum and difference identities express sin(A±B), cos(A±B), and tan(A±B) in terms of sin and cos of the individual angles A and B. They work for all angles and are the foundation for double-angle, half-angle, and product-to-sum formulas.
1. The Six Sum and Difference Formulas
Notice the sign pattern: for sine, the sign on the right side matches the sign on the left (plus gives plus, minus gives minus). For cosine, the signs are opposite (plus gives minus, minus gives plus). The tangent formulas have a denominator that prevents division by zero when tanA·tanB = 1.
2. Worked Example: Finding sin(75°)
To find sin(75°), we decompose 75° as 45° + 30° and apply the sine sum formula:
- Apply the formula: sin(45° + 30°) = sin45°·cos30° + cos45°·sin30°
- Substitute values: (√2/2)(√3/2) + (√2/2)(1/2)
- Multiply: √6/4 + √2/4
- Combine: sin(75°) = (√6 + √2)/4 ≈ 0.9659
This technique works for any angle that can be expressed as a sum or difference of standard angles (0°, 30°, 45°, 60°, 90°).
3. Practical Applications
- Signal Processing: Combining two sinusoidal signals of different frequencies.
- Physics: Analyzing wave interference patterns where two waves combine.
- Navigation: Calculating positions when combining bearing angles.
- Calculus: Integrating products of trigonometric functions using identity transformations.
- Music Theory: Understanding how chords combine frequencies.
4. Frequently Asked Questions
Why do the cosine formulas have opposite signs?
This comes from the unit circle derivation. When you rotate by A+B, the x-coordinate (cosine) involves a subtraction due to the geometry of rotation matrices.
Can I use these for angles greater than 360°?
Yes. The sum and difference identities work for all real numbers, not just acute angles.
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