Finding Exact Trigonometric Values Using Sum And Difference Formulas

Introduction: The Power of Sum and Difference Formulas

In the realm of trigonometry, sum and difference formulas stand as powerful tools, enabling us to simplify complex trigonometric expressions and determine the exact values of trigonometric functions for angles that are not directly available on the unit circle. These formulas are derived from the fundamental trigonometric identities and provide a systematic approach to tackling problems involving the sums or differences of angles. This comprehensive exploration will delve into the intricacies of using the cosine sum formula to find the exact value of the expression cos(7π/10)cos(2π/15) - sin(7π/10)sin(2π/15). We will begin by understanding the underlying principles of sum and difference formulas, specifically focusing on the cosine sum formula. Then, we will apply this knowledge to the given expression, breaking it down step by step to arrive at the final solution. Along the way, we will encounter techniques for converting angles into suitable forms, utilizing trigonometric identities, and simplifying expressions. This journey will equip you with the skills to confidently solve similar trigonometric problems and appreciate the elegance and power of these mathematical tools.

Understanding Sum and Difference Formulas: A Foundation for Trigonometric Problem Solving

The sum and difference formulas in trigonometry are a set of identities that allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. These formulas are essential for simplifying trigonometric expressions, solving equations, and finding exact values of trigonometric functions for angles that are not standard angles (i.e., angles whose trigonometric values are easily memorized or derived from the unit circle, such as 0, π/6, π/4, π/3, and π/2). The sum and difference formulas are derived from the fundamental trigonometric identities and geometric relationships, and they form the basis for many advanced trigonometric concepts. They provide a systematic way to break down complex trigonometric problems into simpler, manageable parts. Among these formulas, the cosine sum formula plays a crucial role in problems involving products of cosine and sine functions, such as the one we are about to explore. Understanding the derivation and application of these formulas is a cornerstone of mastering trigonometry. The ability to recognize patterns and apply the appropriate formula is key to success in this area of mathematics. Moreover, these formulas are not just theoretical constructs; they have practical applications in various fields, including physics, engineering, and computer graphics, where trigonometric functions are used to model periodic phenomena and wave behavior.

Identifying the Relevant Formula: The Cosine Sum Formula

The expression cos(7π/10)cos(2π/15) - sin(7π/10)sin(2π/15) immediately suggests the use of a trigonometric identity involving the product of cosines and sines. The specific structure of this expression, with the product of cosines minus the product of sines, aligns perfectly with the cosine sum formula. This formula states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B), where A and B are any angles. Recognizing this pattern is the crucial first step in solving the problem. The cosine sum formula is one of the fundamental trigonometric identities, and it is essential to have a firm understanding of its structure and application. The formula essentially allows us to condense an expression involving the product of trigonometric functions into a single cosine function of a sum of angles. This simplification is often the key to finding exact values or further simplifying complex trigonometric expressions. In our case, by identifying the correspondence between the given expression and the cosine sum formula, we can rewrite the expression in a more manageable form. The ability to recognize these patterns and apply the appropriate trigonometric identities is a hallmark of proficiency in trigonometry. Furthermore, understanding the derivation of these formulas can enhance one's ability to recall and apply them effectively. By grasping the underlying principles, one can not only solve problems mechanically but also gain a deeper appreciation for the elegance and interconnectedness of trigonometric concepts.

Applying the Cosine Sum Formula: A Step-by-Step Solution

Now that we have identified the cosine sum formula as the relevant tool, let's apply it to the expression cos(7π/10)cos(2π/15) - sin(7π/10)sin(2π/15). By comparing the expression with the formula cos(A + B) = cos(A)cos(B) - sin(A)sin(B), we can see that A = 7π/10 and B = 2π/15. Therefore, we can rewrite the given expression as cos(7π/10 + 2π/15). The next step is to find the sum of the angles 7π/10 and 2π/15. To add these fractions, we need a common denominator. The least common multiple of 10 and 15 is 30. So, we convert the fractions to have a denominator of 30: 7π/10 = (7π * 3)/(10 * 3) = 21π/30 and 2π/15 = (2π * 2)/(15 * 2) = 4π/30. Now we can add the fractions: 21π/30 + 4π/30 = 25π/30. Simplifying this fraction, we get 25π/30 = 5π/6. Thus, our expression simplifies to cos(5π/6). Now, we need to find the exact value of cos(5π/6). This angle is in the second quadrant, where cosine is negative. The reference angle for 5π/6 is π - 5π/6 = π/6. We know that cos(π/6) = √3/2. Since cosine is negative in the second quadrant, cos(5π/6) = -√3/2. Therefore, the exact value of the expression cos(7π/10)cos(2π/15) - sin(7π/10)sin(2π/15) is -√3/2. This step-by-step process demonstrates the power of the cosine sum formula in simplifying complex trigonometric expressions and finding exact values. The ability to manipulate fractions, identify reference angles, and understand the signs of trigonometric functions in different quadrants are all crucial skills in this process.

Conclusion: Mastering Trigonometric Identities for Success

In conclusion, we have successfully used the cosine sum formula to find the exact value of the expression cos(7π/10)cos(2π/15) - sin(7π/10)sin(2π/15). This problem highlights the importance of recognizing patterns in trigonometric expressions and applying the appropriate trigonometric identities. The cosine sum formula, along with other sum and difference formulas, provides a powerful tool for simplifying expressions and solving trigonometric equations. By understanding the underlying principles and practicing the application of these formulas, one can develop a strong foundation in trigonometry. The ability to manipulate trigonometric expressions and find exact values is not only essential for success in mathematics courses but also has applications in various fields, including physics, engineering, and computer science. Mastering trigonometric identities is a key step towards becoming proficient in these areas. The process we followed in this problem—identifying the relevant formula, applying it step-by-step, and simplifying the result—is a general strategy that can be applied to a wide range of trigonometric problems. Continuous practice and a deep understanding of the fundamental concepts are essential for building confidence and expertise in trigonometry. Moreover, exploring the connections between trigonometric identities and other areas of mathematics can further enhance one's appreciation for the beauty and power of these mathematical tools. The journey through trigonometry is a rewarding one, filled with opportunities to discover elegant solutions and deepen one's understanding of the mathematical world.