Find Exact Value Of Sin(140)cos(20) - Cos(140)sin(20)

This article delves into the trigonometric problem of finding the exact value of the expression sin(140°)cos(20°) - cos(140°)sin(20°). We will leverage trigonometric identities, specifically the sine subtraction formula, to simplify the expression and arrive at the solution. This problem not only tests your understanding of trigonometric identities but also highlights the importance of recognizing patterns and applying appropriate formulas to solve seemingly complex problems.

Understanding the Sine Subtraction Formula

At the heart of solving this problem lies the sine subtraction formula, a fundamental identity in trigonometry. This formula states that for any two angles, A and B:

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

This identity is crucial for simplifying expressions involving the sines and cosines of angle differences. By recognizing the pattern in the given expression, we can directly apply this formula to significantly reduce the complexity of the problem. In our case, we have an expression that perfectly matches the right-hand side of the sine subtraction formula, allowing us to rewrite it in a much simpler form.

The sine subtraction formula is derived from geometric considerations and the unit circle definition of trigonometric functions. It's a cornerstone of trigonometric manipulations and appears frequently in various mathematical contexts, including calculus, physics, and engineering. A solid grasp of this formula is essential for anyone working with trigonometric functions. The formula is not just a mathematical tool; it's a powerful concept that reveals the relationship between different trigonometric functions and angles. Its applications extend beyond simple calculations, playing a critical role in fields like signal processing, wave mechanics, and navigation. By mastering this identity, you unlock a deeper understanding of the underlying principles governing trigonometric relationships. The ability to recognize and apply this formula efficiently is a hallmark of a proficient problem-solver in mathematics and related disciplines. Furthermore, the sine subtraction formula serves as a building block for deriving other trigonometric identities, such as the sine addition formula and the double-angle formulas. Its importance in the broader landscape of trigonometry cannot be overstated. Therefore, dedicating time to understanding its origins, applications, and variations is a worthwhile investment for anyone pursuing studies or careers involving mathematics.

Applying the Formula to the Problem

Now, let's apply the sine subtraction formula to our problem. We are given the expression:

sin(140°)cos(20°) - cos(140°)sin(20°)

Comparing this with the sine subtraction formula, sin(A - B) = sin(A)cos(B) - cos(A)sin(B), we can identify that:

• A = 140°
• B = 20°

Therefore, we can rewrite the given expression as:

sin(140° - 20°)

This simplifies to:

sin(120°)

By directly applying the sine subtraction formula, we've transformed a complex-looking expression into a much simpler one. This demonstrates the power of recognizing patterns and utilizing appropriate identities in problem-solving. The key to success here was identifying the structure of the expression as matching the right-hand side of the sine subtraction formula. This allowed us to bypass the need to calculate the individual sine and cosine values of 140° and 20°, which would have been a more cumbersome approach. Instead, we were able to consolidate the expression into a single sine function of a single angle. This highlights the elegance and efficiency that trigonometric identities can bring to mathematical manipulations. The ability to strategically apply these identities is a crucial skill for anyone working in fields that rely on trigonometric calculations. Furthermore, this approach underscores the importance of having a strong foundation in trigonometric identities. Without a solid understanding of these formulas, it would be difficult to recognize the patterns that allow for such simplification. Therefore, mastering trigonometric identities is not just about memorizing formulas; it's about developing the ability to see the underlying structure of mathematical expressions and applying the appropriate tools to simplify them.

Evaluating sin(120°)

To find the exact value of sin(120°), we can use the properties of the unit circle or the sine of supplementary angles. We know that 120° is in the second quadrant, where the sine function is positive. We can express 120° as:

120° = 180° - 60°

Using the property that sin(180° - θ) = sin(θ), we have:

sin(120°) = sin(180° - 60°) = sin(60°)

We know that sin(60°) = √3 / 2. Therefore:

sin(120°) = √3 / 2

This step demonstrates the importance of understanding the relationship between angles and their trigonometric values. By recognizing that 120° is a supplementary angle to 60°, we were able to leverage the known value of sin(60°) to find the value of sin(120°). This approach avoids the need for direct calculation or the use of calculators, highlighting the power of understanding trigonometric properties and relationships. The unit circle provides a visual representation of these relationships, allowing for a deeper understanding of how angles and their trigonometric values are connected. The sine function's behavior in different quadrants is also crucial to grasp, as it determines the sign of the sine value. In the second quadrant, where 120° lies, the sine function is positive, which is why we knew that sin(120°) would be positive. This understanding of quadrant signs is essential for accurate trigonometric calculations. Furthermore, the ability to express angles in terms of their reference angles (in this case, 60°) is a valuable skill that simplifies trigonometric evaluations. By combining these concepts – supplementary angles, quadrant signs, and reference angles – we can efficiently and accurately determine the values of trigonometric functions for a wide range of angles.

The Final Answer

Therefore, the exact value of sin(140°)cos(20°) - cos(140°)sin(20°) is √3 / 2. Looking at the options provided:

A. -1/2 B. -√3 / 2 C. √3 / 2 D. 1/2

The correct answer is C. √3 / 2. This solution demonstrates the power of trigonometric identities in simplifying complex expressions. By applying the sine subtraction formula, we were able to reduce the problem to a straightforward evaluation of sin(120°). This not only provides the correct answer but also illustrates a key problem-solving strategy in trigonometry: recognizing patterns and applying appropriate formulas to simplify expressions.

This problem serves as a valuable example of how trigonometric identities can be used to solve seemingly complicated problems in an elegant and efficient manner. The ability to recognize patterns, apply the correct identities, and simplify expressions is a crucial skill for anyone working with trigonometry. Moreover, this problem underscores the importance of having a strong foundation in trigonometric fundamentals. Without a thorough understanding of identities like the sine subtraction formula, it would be difficult to approach this problem effectively. The solution also highlights the interconnectedness of trigonometric concepts. The ability to relate angles, use reference angles, and understand the behavior of trigonometric functions in different quadrants are all essential components of a comprehensive understanding of trigonometry. Therefore, mastering these fundamentals is not just about memorizing formulas; it's about developing a deep understanding of the underlying principles that govern trigonometric relationships. This deeper understanding allows for greater flexibility in problem-solving and a more intuitive grasp of trigonometric concepts. In conclusion, the solution to this problem demonstrates the power of trigonometric identities and the importance of a solid foundation in trigonometric fundamentals.