Calculating trigonometric values for angles that aren't standard ones like 0°, 30°, 45°, 60°, and 90° often requires the use of trigonometric identities. In this article, we will delve into determining the exact value of cos(-15°). This exploration will not only solidify your understanding of trigonometric principles but also showcase the elegance and power of these mathematical tools. To find the exact value of cos(-15°), we will use a combination of trigonometric identities, including the cosine of a difference formula and the properties of cosine for negative angles. This process will highlight the importance of understanding and applying these concepts effectively in solving trigonometric problems. The goal is to break down the problem into manageable parts, utilize the known values of standard angles, and arrive at the precise value of cos(-15°).
Understanding the Problem
Before diving into the solution, let's first understand the problem at hand. We are tasked with finding the exact value of the cosine of -15 degrees, or cos(-15°). This angle is not a standard angle for which we typically memorize trigonometric values. Therefore, we need to use trigonometric identities to express -15° in terms of angles whose trigonometric values we know. Common strategies include using the sum or difference formulas for cosine and sine, or employing half-angle formulas. In this case, we can express -15° as a combination of standard angles like 45° and 60° or 30° and 45°. The key here is to recognize that -15° can be written as 45° - 60° or 30° - 45°, both of which will lead us to a solution using the cosine difference formula. Additionally, understanding that cosine is an even function, meaning cos(-x) = cos(x), allows us to simplify the problem further. This property will be particularly useful in our calculations, ensuring that we handle the negative angle correctly. By clearly defining the problem and considering the available strategies, we set the stage for a systematic and accurate solution. The initial understanding of the problem and the recognition of applicable trigonometric identities are crucial steps in solving this type of question.
Trigonometric Identities
To solve this problem, we need to employ some key trigonometric identities. One of the most crucial identities for finding cos(-15°) is the cosine difference formula, which states:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
This formula allows us to express the cosine of the difference of two angles in terms of the cosines and sines of the individual angles. In our case, we can use this formula by expressing -15° as the difference of two angles whose trigonometric values are well-known, such as 45° and 60°. Another important concept is the even property of the cosine function. This property tells us that the cosine of a negative angle is the same as the cosine of its positive counterpart. Mathematically, this is expressed as:
cos(-x) = cos(x)
This property simplifies our problem because it allows us to rewrite cos(-15°) as cos(15°). By using these identities, we can break down the problem into smaller, more manageable parts. We will substitute the appropriate angles into the cosine difference formula and then use the known values of the trigonometric functions for these standard angles to find the exact value of cos(-15°). Understanding and applying these trigonometric identities correctly is essential for solving this problem accurately and efficiently. These formulas provide the necessary tools to transform the problem into a solvable form.
Applying the Cosine Difference Formula
Now, let's apply the cosine difference formula to find the exact value of cos(-15°). First, we use the property that cos(-x) = cos(x) to rewrite cos(-15°) as cos(15°). Next, we express 15° as the difference between two standard angles, 45° and 30°:
15° = 45° - 30°
Now we can use the cosine difference formula:
cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°)
We know the exact values of the trigonometric functions for 45° and 30°:
cos(45°) = \frac{\sqrt{2}}{2}
sin(45°) = \frac{\sqrt{2}}{2}
cos(30°) = \frac{\sqrt{3}}{2}
sin(30°) = \frac{1}{2}
Substituting these values into the formula, we get:
cos(15°) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)
Simplifying the expression, we have:
cos(15°) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
Combining the terms, we obtain:
cos(15°) = \frac{\sqrt{6} + \sqrt{2}}{4}
Thus, the exact value of cos(-15°) is the same as cos(15°), which is (√6 + √2)/4. This step-by-step application of the cosine difference formula, combined with accurate substitution of known trigonometric values, leads us to the final solution. The careful execution of each step ensures the correctness of the result. This process demonstrates the power and utility of trigonometric identities in solving problems involving non-standard angles.
Final Answer
Therefore, the exact value of cos(-15°) is:
\frac{\sqrt{6} + \sqrt{2}}{4}
This result matches option D, which is the correct answer. In summary, to find the exact value of cos(-15°), we used the property that cosine is an even function, allowing us to rewrite cos(-15°) as cos(15°). Then, we expressed 15° as the difference between 45° and 30° and applied the cosine difference formula. By substituting the known values of the trigonometric functions for 45° and 30°, we simplified the expression and arrived at the final answer. This process highlights the importance of understanding and applying trigonometric identities in solving problems involving non-standard angles. The use of these identities allows us to break down complex problems into simpler parts, making them more manageable and solvable. The final answer, (√6 + √2)/4, is a precise value that demonstrates the elegance and accuracy of trigonometric calculations. This detailed solution provides a comprehensive understanding of how to tackle similar problems in the future.
Conclusion
In conclusion, we have successfully determined the exact value of cos(-15°) by leveraging key trigonometric identities and principles. This exercise underscores the importance of understanding and applying these concepts in solving trigonometric problems. The use of the cosine difference formula and the even property of the cosine function were crucial in simplifying the problem and arriving at the accurate solution. The ability to express non-standard angles in terms of standard angles allows us to calculate their trigonometric values precisely. Throughout this process, we emphasized the step-by-step approach, ensuring clarity and accuracy in our calculations. The final result, (√6 + √2)/4, exemplifies the elegance and precision of trigonometric mathematics. This method can be applied to find the exact values of other non-standard angles as well, reinforcing the versatility and power of trigonometric identities. Mastering these techniques not only enhances problem-solving skills but also deepens the understanding of the fundamental principles underlying trigonometry. This comprehensive exploration provides a solid foundation for tackling more complex trigonometric problems in the future, further solidifying the importance of a strong grasp of these foundational concepts.
