Simplifying Trigonometric Expressions An In-Depth Guide

In the realm of trigonometry, mathematical expressions often present themselves in forms that require simplification or transformation to reveal their true nature. One such expression is $\cos (103^{\circ}) \cos (54^{\circ})-\sin (103^{\circ}) \sin (54^{\circ})$. This expression, at first glance, might seem complex, but it elegantly embodies a fundamental trigonometric identity. In this comprehensive exploration, we will dissect this expression, unveil the underlying trigonometric principle, and determine its equivalent form from the given options. This journey will not only provide the answer but also deepen your understanding of trigonometric identities and their applications. We will navigate through the intricacies of angle addition and subtraction formulas, highlighting their significance in simplifying complex expressions. The aim is to equip you with the knowledge and skills to confidently tackle similar problems in trigonometry, fostering a robust understanding of the subject matter. This in-depth analysis will serve as a valuable resource for students, educators, and anyone with a keen interest in mathematics.

Understanding the Core Trigonometric Identity

The key to simplifying the given trigonometric expression lies in recognizing a specific trigonometric identity: the cosine angle addition formula. This formula states that for any two angles, A and B, the cosine of their sum is given by:

$\cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B)$

This identity is a cornerstone of trigonometry, allowing us to express the cosine of a sum of angles in terms of the cosines and sines of the individual angles. It's a powerful tool for simplifying expressions and solving trigonometric equations. The beauty of this identity lies in its ability to transform a seemingly complex expression into a more manageable form. By recognizing this pattern, we can directly apply the formula to our problem, making the simplification process straightforward and efficient. This section will delve deeper into the origins and applications of this identity, providing a solid foundation for understanding its importance in trigonometry. We will also explore how this identity relates to other trigonometric identities, such as the sine angle addition formula and the angle subtraction formulas.

Applying the Identity to the Given Expression

Now, let's apply this fundamental trigonometric identity to the given expression: $\cos (103^{\circ}) \cos (54^{\circ})-\sin (103^{\circ}) \sin (54^{\circ})$. By comparing this expression with the cosine angle addition formula, we can see a direct correspondence. We can identify A as $103^{\circ}$ and B as $54^{\circ}$. Therefore, the expression can be rewritten as:

$\cos (103^{\circ}) \cos (54^{\circ})-\sin (103^{\circ}) \sin (54^{\circ}) = \cos (103^{\circ} + 54^{\circ})$

This step is crucial as it transforms the product of cosines and sines into a single cosine function, significantly simplifying the expression. This transformation highlights the power of trigonometric identities in simplifying complex expressions. The next step involves a simple arithmetic operation, adding the two angles together. This section will further elaborate on the importance of pattern recognition in mathematics and how it can lead to efficient problem-solving. We will also discuss common pitfalls to avoid when applying trigonometric identities.

Simplifying the Angle Sum

The next step is to calculate the sum of the angles: $103^{\circ} + 54^{\circ}$. This is a straightforward addition:

$103^{\circ} + 54^{\circ} = 157^{\circ}$

Therefore, our expression now becomes:

$\cos (103^{\circ} + 54^{\circ}) = \cos (157^{\circ})$

This simplification brings us closer to the solution. We have successfully reduced the original complex expression to a single cosine function of a specific angle. This step underscores the importance of basic arithmetic skills in conjunction with trigonometric knowledge. The ability to perform simple calculations accurately is essential for solving mathematical problems efficiently. This section will also briefly touch upon the concept of angle measurement and different units of angle measurement, such as degrees and radians.

Identifying the Equivalent Expression

Comparing our simplified expression $\cos (157^{\circ})$ with the given options:

A. $\sin (49^{\circ})$ B. $\cos (49^{\circ})$ C. $\sin (157^{\circ})$ D. $\cos (157^{\circ})$

We can clearly see that option D, $\cos (157^{\circ})$, matches our simplified expression. Therefore, the equivalent expression is $\cos (157^{\circ})$. This final step emphasizes the importance of careful comparison and attention to detail. Ensuring that the simplified expression aligns perfectly with one of the given options is crucial for arriving at the correct answer. This section will also briefly discuss the concept of equivalent expressions in mathematics and how different expressions can represent the same mathematical value.

Conclusion: The Power of Trigonometric Identities

In conclusion, the expression $\cos (103^{\circ}) \cos (54^{\circ})-\sin (103^{\circ}) \sin (54^{\circ})$ is equivalent to $\cos (157^{\circ})$. This simplification was achieved by recognizing and applying the cosine angle addition formula, a fundamental trigonometric identity. This exercise highlights the power of trigonometric identities in simplifying complex expressions and revealing their underlying simplicity. Mastering these identities is crucial for success in trigonometry and related fields. The ability to recognize patterns, apply appropriate formulas, and simplify expressions is a valuable skill in mathematics and beyond. This exploration serves as a testament to the elegance and efficiency of trigonometric principles. By understanding and applying these principles, we can unravel complex mathematical problems and arrive at concise and accurate solutions. This journey through trigonometric simplification reinforces the importance of a strong foundation in mathematical concepts and the ability to apply them creatively and effectively. We encourage further exploration of trigonometric identities and their applications in various mathematical contexts.