Steps to Solve Problems with Sum-to-Product Identities in Trigonometry

📚 Introduction to Sum-to-Product Identities

Sum-to-product identities are a set of trigonometric formulas that allow us to express sums or differences of trigonometric functions as products. These identities are incredibly useful in simplifying expressions, solving trigonometric equations, and proving other trigonometric identities. They're derived from the angle addition and subtraction formulas.

📜 History and Background

The development of trigonometric identities, including sum-to-product formulas, can be traced back to ancient Greek mathematicians like Hipparchus and Ptolemy, who created early trigonometric tables for astronomical calculations. Over centuries, mathematicians from India and the Islamic world further refined these concepts. François Viète, a 16th-century French mathematician, significantly contributed to the symbolic representation of trigonometric functions, laying the groundwork for the identities we use today. These identities became essential tools in fields such as navigation, surveying, and physics.

🔑 Key Principles and Formulas

• ➕Sum of Sines: $\sin(x) + \sin(y) = 2 \sin(\frac{x+y}{2}) \cos(\frac{x-y}{2})$
• ➖Difference of Sines: $\sin(x) - \sin(y) = 2 \cos(\frac{x+y}{2}) \sin(\frac{x-y}{2})$
• ➕Sum of Cosines: $\cos(x) + \cos(y) = 2 \cos(\frac{x+y}{2}) \cos(\frac{x-y}{2})$
• ➖Difference of Cosines: $\cos(x) - \cos(y) = -2 \sin(\frac{x+y}{2}) \sin(\frac{x-y}{2})$

📝 Steps to Solve Problems

• 🔍 Identify the Pattern: Determine if the problem involves a sum or difference of sines or cosines.
• ✍️ Apply the Correct Identity: Choose the appropriate sum-to-product identity based on the identified pattern.
• 🔢 Substitute Values: Plug in the given values for $x$ and $y$ into the identity.
• 🧮 Simplify: Simplify the expression using algebraic and trigonometric manipulations.
• ✅ Verify: Check your answer for correctness, ensuring it makes sense in the context of the problem.

💡 Real-World Examples

Example 1: Simplify $\sin(75^\circ) + \sin(15^\circ)$.

Solution: Using the sum of sines identity:

$\sin(75^\circ) + \sin(15^\circ) = 2 \sin(\frac{75^\circ + 15^\circ}{2}) \cos(\frac{75^\circ - 15^\circ}{2}) = 2 \sin(45^\circ) \cos(30^\circ) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}$

Example 2: Simplify $\cos(5x) - \cos(3x)$.

Solution: Using the difference of cosines identity:

$\cos(5x) - \cos(3x) = -2 \sin(\frac{5x + 3x}{2}) \sin(\frac{5x - 3x}{2}) = -2 \sin(4x) \sin(x)$

✍️ Practice Quiz

Try these problems to test your understanding:

1. Simplify: $\sin(50^\circ) + \sin(10^\circ)$
2. Simplify: $\cos(80^\circ) + \cos(40^\circ)$
3. Simplify: $\sin(6x) - \sin(2x)$
4. Simplify: $\cos(7x) - \cos(x)$
5. Evaluate: $\frac{\sin(5x) + \sin(3x)}{\cos(5x) + \cos(3x)}$

Solutions:

1. $\sqrt{3}/2$
2. $\cos(60^\circ)$
3. $2 \cos(4x) \sin(2x)$
4. $-2 \sin(4x) \sin(3x)$
5. $\tan(4x)$

🎓 Conclusion

Sum-to-product identities are powerful tools in trigonometry. By understanding the formulas and practicing their application, you can simplify complex expressions and solve a wide range of trigonometric problems. Keep practicing, and you'll master these identities in no time!