Simplifying Trigonometric Expressions To A Single Angle And Evaluation

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Hey guys! Let's dive into the fascinating world of trigonometry and tackle a cool problem together. We're going to take a trigonometric expression and simplify it into a single angle, and then evaluate it. Sounds fun, right? So, let's get started and break it down step by step. Our mission is to rewrite the expression $\sin 20^{\circ} \cos 16^{\circ} + \cos 20^{\circ} \sin 16^{\circ}$ as a trigonometric function of a single angle. This involves recognizing a fundamental trigonometric identity that will help us simplify the expression. Understanding these identities is crucial not just for solving problems, but for truly grasping the essence of how trigonometric functions behave and interact. Think of it like learning a new language; once you understand the grammar (the identities), you can start to form sentences (solve problems) with ease and elegance.

The Sine Addition Formula: The cornerstone of our solution is the sine addition formula. This formula is a powerful tool in trigonometry, allowing us to combine sine and cosine functions of different angles into a single sine function. The sine addition formula states that for any angles A and B:

sin(A+B)=sinAcosB+cosAsinB\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B

This identity is not just a random equation; it’s a fundamental relationship that arises from the geometry of the unit circle and the definitions of sine and cosine. It’s like a secret key that unlocks many trigonometric puzzles. By recognizing this pattern, we can significantly simplify complex expressions and make them much easier to work with. The beauty of this formula lies in its ability to condense two separate trigonometric terms into one, making subsequent calculations and analyses much more straightforward. In practical applications, this can be incredibly useful, such as in physics when dealing with wave phenomena or in engineering when analyzing oscillating systems. Remembering this formula is crucial, but understanding its derivation and the logic behind it can make it even more memorable and easier to apply in various contexts.

Applying the Formula: Now, let’s put this formula to work! When we look at our expression, $\sin 20^{\circ} \cos 16^{\circ} + \cos 20^{\circ} \sin 16^{\circ}$, we can see that it perfectly matches the right-hand side of the sine addition formula. It’s like finding a missing piece in a jigsaw puzzle. We have a clear correspondence:

  • A = 20°
  • B = 16°

This is where the magic happens. We can directly substitute these values into the left-hand side of the formula. It’s a moment of clarity when you realize how neatly the pieces fit together. This direct application of the formula transforms our complex expression into something much simpler and more manageable. It's not just about plugging in numbers; it’s about recognizing the underlying structure and leveraging a known identity to simplify the problem. This step highlights the power of pattern recognition in mathematics and the efficiency that comes with using established formulas and identities. By seeing the expression in terms of the formula, we’re essentially translating it into a different language, one that allows for easier manipulation and solution.

Rewriting the Expression: By applying the sine addition formula, we can rewrite our original expression as:

sin(20∘+16∘)\\sin(20^{\circ} + 16^{\circ})

This is a significant step forward. We’ve taken a sum of two products and condensed it into a single sine function. It’s like turning a complicated recipe with multiple ingredients into a simple, one-dish meal. This simplification not only makes the expression easier to handle but also provides a clearer understanding of its underlying nature. We've gone from a somewhat convoluted form to a concise and elegant representation. The beauty of this transformation lies in its simplicity. It demonstrates how a well-chosen identity can dramatically reduce the complexity of a problem, making it much more accessible and easier to solve. In essence, we’ve used a mathematical tool to streamline our expression, setting the stage for the next step: evaluation.

Adding the Angles: The next step is straightforward but crucial. We simply add the angles inside the sine function:

sin(20∘+16∘)=sin(36∘)\\sin(20^{\circ} + 16^{\circ}) = \\sin(36^{\circ})

This step is like combining the final ingredients in our simplified recipe. We’ve reduced the problem to a single trigonometric function of a single angle. This is a significant achievement, as it allows us to directly evaluate the expression. The arithmetic here is simple, but the impact is profound. By performing this addition, we’ve transformed the expression into its most basic form, ready for final evaluation. This step highlights the importance of methodical simplification in mathematics. Each step, no matter how small, contributes to the overall solution. The sum of the angles gives us a clear, single angle to work with, making the final calculation much easier and more direct.

Evaluating the Sine Function: Now, we need to evaluate $\sin(36^{\circ})$. This is where things get a little trickier, as 36° is not one of the standard angles (0°, 30°, 45°, 60°, 90°) for which we typically memorize sine values. So, what do we do? We have a couple of options:

  • Using a Calculator: The most direct way is to use a calculator. Make sure your calculator is in degree mode, and simply input $\sin(36)$. The calculator will give you an approximate decimal value.
  • Looking Up a Table: You can also refer to a trigonometric table, which lists the values of sine, cosine, and tangent for various angles. These tables were commonly used before the widespread availability of calculators and are still a valuable resource for precise values.
  • Special Angles and Identities: For those who love a mathematical challenge, it’s worth noting that $\sin(36^{\circ})$ can be expressed in terms of radicals using some clever geometry and the properties of the golden ratio. However, this approach is more complex and typically not necessary for most problem-solving scenarios.

For our purposes, using a calculator is the most efficient method. When you input $\sin(36)$, you’ll get a value that’s approximately 0.5878. This numerical value is the culmination of all our previous steps. It’s the final answer, the result of transforming and simplifying the original expression. The use of a calculator here is not just about getting a number; it’s about bridging the gap between symbolic manipulation and numerical evaluation. It’s a practical step that allows us to express the result in a tangible, usable form.

The Final Answer: Therefore, $\sin 20^{\circ} \cos 16^{\circ} + \cos 20^{\circ} \sin 16^{\circ} = \sin(36^{\circ}) \approx 0.5878$. We’ve successfully rewritten the expression as a trigonometric function of a single angle and evaluated it! It’s like reaching the summit after a challenging climb. We’ve started with a complex expression, navigated through the terrain of trigonometric identities, and arrived at a clear, concise answer. This final result is not just a number; it’s a testament to the power of mathematical reasoning and the elegance of trigonometric relationships. It demonstrates how seemingly complicated problems can be broken down into manageable steps, leading to a satisfying and concrete solution.

Let’s recap what we’ve done and highlight the key takeaways from this exercise. This is like summarizing the main points of a story, ensuring that we’ve captured the essence of our journey through the problem.

  1. Recognizing Trigonometric Identities: The first and perhaps most crucial step is recognizing the sine addition formula. This is like having the right tool for the job. Without it, the problem would be much harder to solve. The ability to identify and apply trigonometric identities is a fundamental skill in mathematics. It’s not just about memorization; it’s about understanding the patterns and relationships that these identities represent. Recognizing these patterns allows you to see connections between different parts of a problem and to choose the most efficient path to a solution.

  2. Applying the Formula: Once we identified the formula, we applied it directly to rewrite the expression. This is like using the tool to perform the specific task. The direct application of the sine addition formula demonstrates the power of these identities in simplifying complex expressions. It’s a clear example of how a well-chosen identity can transform a problem into a more manageable form. This step highlights the importance of being able to translate a problem into a form where known mathematical tools can be applied effectively.

  3. Simplifying and Evaluating: We then simplified the expression by adding the angles and evaluated the sine function using a calculator. This is like putting the finishing touches on a masterpiece. The simplification and evaluation steps bring the problem to a clear, numerical conclusion. These steps demonstrate the importance of precision and attention to detail in mathematics. Each calculation, no matter how small, contributes to the overall accuracy of the solution. The use of a calculator in the final evaluation underscores the practical aspect of mathematics, where symbolic manipulation leads to concrete numerical results.

This problem beautifully illustrates how trigonometric identities can simplify complex expressions and lead to elegant solutions. It’s not just about getting the right answer; it’s about the journey of understanding and applying mathematical principles. So, keep practicing, keep exploring, and you’ll become a trigonometry whiz in no time!

Trigonometric Simplification: Solving sin 20° cos 16° + cos 20° sin 16°