The Sum and Difference Formulas (also known as addition and subtraction identities) are powerful tools in trigonometry. They allow you to rewrite trigonometric functions involving sums (like $A + B$) or differences (like $A - B$) into expressions involving only sine and cosine of $A$ and $B$.
[Image of sum and difference formulas trigonometry]Why do we need them? While we memorize values for standard angles like 30°, 45°, and 60°, what happens if we need the exact sine of 75°? Using these formulas, we can break 75° into (30° + 45°) and solve it precisely without a calculator.
1. The Sine Formulas
The sine formulas are friendly because they "keep the sign" (plus stays plus, minus stays minus) but they mix the functions (sine goes with cosine).
[Image of sine sum formula geometric proof]sin(A - B) = sinA cosB - cosA sinB
Mnemonic: "Sine mixes signs but keeps the symbol."
2. The Cosine Formulas
The cosine formulas are the "stubborn" ones. They group functions together (cosines with cosines, sines with sines) but they flip the sign (plus becomes minus).
cos(A - B) = cosA cosB + sinA sinB
Mnemonic: "Cosine is cliquey (stays with its own kind) and contrary (flips the sign)."
3. The Tangent Formulas
The tangent formulas are derived by dividing the sine formula by the cosine formula.
tan(A - B) = (tanA - tanB) / (1 + tanA tanB)
Notice that the sign on top matches the operation, while the sign on the bottom is opposite.
4. Practical Example: Finding Exact Values
Let's find the exact value of cos(15°).
Since 15° isn't on the standard unit circle, we can rewrite it as a difference of two angles we do know: 45° - 30°.
[Image of calculating sin 75 degrees using sum formula]- Identify the formula: We need $cos(A - B)$.
- Set variables: $A = 45°$, $B = 30°$.
- Substitute: $cos(45°-30°) = cos(45°)cos(30°) + sin(45°)sin(30°)$.
- Evaluate:
$= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2})$
$= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$ - Final Answer: $\frac{\sqrt{6} + \sqrt{2}}{4}$
5. Proof and Origins
These formulas aren't magic; they are derived from geometry. One common proof involves the distance formula on the unit circle or geometric construction using triangles. They also relate deeply to Euler's Formula ($e^{ix} = \cos x + i\sin x$), which provides the most elegant proof in higher mathematics.
Conclusion
The Sum and Difference Formulas expand your trigonometric toolkit significantly. They are the bridge that allows you to calculate new angles from old ones and are essential for simplifying complex wave equations in physics and engineering.