Finding The Exact Value Of Sin(264°)cos(6°) + Cos(264°)sin(6°)
This article delves into the fascinating world of trigonometry to solve a seemingly complex problem: finding the exact value of the expression sin(264°)cos(6°) + cos(264°)sin(6°). This expression, at first glance, might appear daunting. But with the application of a fundamental trigonometric identity and a bit of algebraic manipulation, we can unravel its true value. Our journey will take us through the application of the sine addition formula, understanding reference angles, and ultimately arriving at a simplified, exact answer. This exploration highlights the elegance and power of trigonometric principles in simplifying mathematical expressions and revealing hidden relationships between angles and their trigonometric functions.
Leveraging the Sine Addition Formula
The sine addition formula is a cornerstone of trigonometry, providing a powerful tool for simplifying expressions involving the sines and cosines of sums of angles. This formula states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Recognizing the pattern in our given expression, sin(264°)cos(6°) + cos(264°)sin(6°), we can directly apply this formula. By substituting A = 264° and B = 6°, we transform the expression into sin(264° + 6°). This simplification is a crucial first step, as it condenses the original expression into a single trigonometric function of a single angle. The ability to recognize and apply trigonometric identities like the sine addition formula is essential for solving a wide range of trigonometric problems, from simplifying expressions to solving equations. Mastering these identities unlocks a deeper understanding of the relationships between angles and their trigonometric functions, paving the way for more advanced mathematical explorations. This foundational knowledge allows us to approach complex problems with confidence, knowing that the right tools are at our disposal.
Simplifying to sin(270°)
Following the application of the sine addition formula, we've simplified our expression to sin(270°). This step is significant because it transforms a complex expression into a single, readily evaluated trigonometric function. Now, the problem boils down to determining the value of sin(270°). To do this, we need to understand the unit circle and how angles are represented within it. The unit circle provides a visual and intuitive way to understand the values of trigonometric functions for any angle. An angle of 270° corresponds to a point on the unit circle directly below the origin, on the negative y-axis. At this point, the coordinates are (0, -1). Since the sine function corresponds to the y-coordinate on the unit circle, sin(270°) is equal to -1. This direct correspondence between angles on the unit circle and the values of their trigonometric functions makes the unit circle an invaluable tool for trigonometric calculations and visualizations. By understanding this relationship, we can quickly determine the values of trigonometric functions for key angles, such as 0°, 90°, 180°, 270°, and 360°.
The Exact Value: -1
Having simplified the expression to sin(270°), we can now definitively state the exact value. As discussed in the previous section, the sine function corresponds to the y-coordinate on the unit circle. For an angle of 270°, the point on the unit circle is (0, -1). Therefore, sin(270°) = -1. This result elegantly concludes our journey, providing a precise and numerical answer to the initial problem. The entire process, from the initial complex expression to the final simple answer, showcases the power of trigonometric identities and the unit circle in simplifying mathematical problems. This journey not only provides the answer but also reinforces the importance of understanding fundamental trigonometric concepts. The ability to manipulate trigonometric expressions, apply identities, and utilize the unit circle are crucial skills for anyone studying mathematics, physics, or engineering. This example serves as a reminder that even seemingly complex problems can be solved with the right tools and a solid understanding of the underlying principles.
Visualizing on the Unit Circle
To solidify our understanding, let's visualize the problem on the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. An angle of 264° lies in the third quadrant, and an angle of 6° is a small angle in the first quadrant. The sine and cosine functions can be interpreted as the y and x coordinates, respectively, of a point on the unit circle corresponding to a given angle. When we add 264° and 6°, we get 270°, which corresponds to the point (0, -1) on the unit circle. This point lies directly on the negative y-axis. The y-coordinate of this point, which represents the sine of 270°, is -1. This visual representation reinforces the connection between angles, the unit circle, and the values of trigonometric functions. By visualizing angles and their corresponding points on the unit circle, we can gain a deeper intuition for the behavior of trigonometric functions. This visual approach is particularly helpful for understanding the signs of trigonometric functions in different quadrants and for quickly determining the values of trigonometric functions for key angles.
Alternative Approaches and Generalizations
While we've successfully solved the problem using the sine addition formula and the unit circle, it's worth considering alternative approaches and generalizations. For instance, one could explore the use of other trigonometric identities, such as the cosine subtraction formula, to arrive at the same solution. Although the sine addition formula provided a direct route, exploring alternative approaches can deepen our understanding of the interconnectedness of trigonometric identities. Furthermore, this problem serves as a stepping stone for more general problems involving the simplification of trigonometric expressions. The techniques we've employed can be applied to a wide range of similar problems, highlighting the versatility of these methods. By mastering these techniques, we can confidently tackle more complex trigonometric challenges. Additionally, understanding the underlying principles allows us to generalize these concepts to other areas of mathematics and physics where trigonometric functions play a crucial role. The ability to adapt and apply these techniques to new situations is a hallmark of a strong mathematical foundation.
Conclusion: The Power of Trigonometric Identities
In conclusion, we've successfully determined the exact value of sin(264°)cos(6°) + cos(264°)sin(6°) to be -1. This journey has showcased the power and elegance of trigonometric identities, particularly the sine addition formula. By applying this formula, we transformed a complex expression into a simple one, sin(270°), which we then evaluated using our knowledge of the unit circle. The unit circle provided a visual and intuitive way to understand the value of sin(270°), reinforcing the connection between angles and their trigonometric functions. This problem serves as a valuable example of how trigonometric identities can simplify mathematical expressions and reveal hidden relationships. The ability to recognize patterns, apply identities, and visualize angles on the unit circle are crucial skills for anyone studying mathematics, physics, or engineering. Mastering these skills empowers us to tackle a wide range of trigonometric problems with confidence and precision. The journey from the initial complex expression to the final simple answer highlights the beauty and power of mathematical reasoning.
