Solving Cos(2x - 10°) ⋅ Sec(x + 30°) = 1 A Step-by-Step Guide

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Introduction

In this comprehensive article, we will delve into the intricacies of solving the trigonometric equation cos(2x - 10°) ⋅ sec(x + 30°) = 1. Trigonometric equations often present a unique set of challenges, requiring a strong understanding of trigonometric identities, algebraic manipulation, and problem-solving strategies. This article aims to provide a step-by-step guide to solving this particular equation, explaining the underlying concepts and techniques involved. Whether you are a student learning trigonometry or someone looking to brush up on your skills, this article will offer valuable insights and a clear methodology for tackling similar problems.

We will begin by transforming the equation into a more manageable form using trigonometric identities. Specifically, we will leverage the reciprocal relationship between secant and cosine. Then, we will explore the conditions under which the equation holds true, leading to a set of possible solutions. These solutions will be further analyzed to ensure they fall within the domain of the original equation, accounting for any restrictions imposed by the trigonometric functions involved. Finally, we will present the general solutions, which encompass all possible values of x that satisfy the given equation. This structured approach will not only help in solving this specific equation but also equip you with a general framework for handling a wide range of trigonometric problems. Understanding the nuances of trigonometric equations is crucial in various fields, including physics, engineering, and mathematics, making this topic an essential part of mathematical education.

Understanding the Trigonometric Equation

To effectively solve the trigonometric equation cos(2x - 10°) ⋅ sec(x + 30°) = 1, it's crucial to first understand the fundamental concepts and relationships involved. This equation combines cosine and secant functions, which are reciprocally related. The cosine function, denoted as cos(θ), represents the x-coordinate of a point on the unit circle corresponding to an angle θ. Its values range from -1 to 1. The secant function, denoted as sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Consequently, secant is undefined where cosine is zero, and its values can be any real number except those between -1 and 1.

The equation cos(2x - 10°) ⋅ sec(x + 30°) = 1 essentially states that the product of the cosine of the angle (2x - 10°) and the secant of the angle (x + 30°) is equal to 1. This relationship can only hold true under specific conditions, which we will explore in detail. One critical aspect to consider is the domain of the functions involved. Since secant is the reciprocal of cosine, sec(x + 30°) is undefined when cos(x + 30°) = 0. This restriction must be taken into account when determining the solutions. Furthermore, understanding the periodicity of trigonometric functions is essential. Both cosine and secant are periodic functions, meaning their values repeat at regular intervals. For cosine, the period is 360°, while for secant, it is also 360°. This periodicity implies that there are infinitely many solutions to trigonometric equations, and we often seek general solutions that express all possible values.

Trigonometric identities play a pivotal role in simplifying and solving trigonometric equations. These identities provide relationships between different trigonometric functions and angles, allowing us to rewrite equations in more manageable forms. In this case, we will use the reciprocal identity sec(θ) = 1/cos(θ) to transform the equation. Additionally, understanding angle manipulation and algebraic techniques is necessary to isolate the variable x and find its values. Solving trigonometric equations requires a combination of trigonometric knowledge, algebraic skills, and careful attention to detail. By mastering these concepts, you can effectively tackle a wide range of trigonometric problems and gain a deeper appreciation for the beauty and power of trigonometry.

Solving the Equation Step-by-Step

To solve the trigonometric equation cos(2x - 10°) ⋅ sec(x + 30°) = 1, we will follow a structured approach that involves transforming the equation, identifying conditions for solutions, and finding the general solutions. Here’s a step-by-step breakdown:

  1. Transform the Equation:

    The first step involves using the reciprocal identity to rewrite the equation in terms of cosine. Recall that sec(θ) = 1/cos(θ). Applying this identity, we can rewrite the given equation as:

    cos(2x - 10°) ⋅ [1/cos(x + 30°)] = 1

    This transformation simplifies the equation and allows us to work with a single trigonometric function, making it easier to manipulate and solve.

  2. Multiply and Simplify:

    Next, we multiply both sides of the equation by cos(x + 30°) to eliminate the fraction. This gives us:

    cos(2x - 10°) = cos(x + 30°)

    Now, the equation is in a form where we can directly compare the cosine values of two different angles.

  3. Identify Conditions for Solutions:

    For the equation cos(A) = cos(B) to hold true, there are two possible conditions:

    • Case 1: A = B + 360°k, where k is an integer. This represents the scenario where the angles are coterminal, differing by a multiple of 360°.
    • Case 2: A = -B + 360°k, where k is an integer. This accounts for the symmetry of the cosine function, where cos(θ) = cos(-θ).

    We will apply these conditions to our equation to find the possible values of x.

  4. Apply Case 1:

    Using the first condition, we set 2x - 10° equal to x + 30° plus a multiple of 360°:

    2x - 10° = x + 30° + 360°k

    Now, we solve for x:

    2x - x = 30° + 10° + 360°k

    x = 40° + 360°k

    This gives us a set of solutions for x based on the integer k.

  5. Apply Case 2:

    Using the second condition, we set 2x - 10° equal to the negative of x + 30° plus a multiple of 360°:

    2x - 10° = -(x + 30°) + 360°k

    2x - 10° = -x - 30° + 360°k

    Now, we solve for x:

    2x + x = -30° + 10° + 360°k

    3x = -20° + 360°k

    x = (-20°/3) + 120°k

    This gives us another set of solutions for x based on the integer k.

  6. Check for Extraneous Solutions:

    Before finalizing our solutions, we need to check for extraneous solutions that may arise due to the original secant function. Recall that sec(x + 30°) is undefined when cos(x + 30°) = 0. So, we need to ensure that our solutions do not make the denominator zero.

    The condition cos(x + 30°) = 0 occurs when x + 30° = 90° + 180°n, where n is an integer. Solving for x, we get:

    x = 60° + 180°n

    We need to make sure that our solutions x = 40° + 360°k and x = (-20°/3) + 120°k do not fall into this category. By inspection, we can see that none of the general solutions will make the original equation undefined, as they do not coincide with the form x = 60° + 180°n.

  7. General Solutions:

    Therefore, the general solutions for the equation cos(2x - 10°) ⋅ sec(x + 30°) = 1 are:

    • x = 40° + 360°k
    • x = (-20°/3) + 120°k

    where k is an integer. These solutions represent all possible values of x that satisfy the given trigonometric equation.

By following these steps, we have successfully solved the trigonometric equation and obtained the general solutions. This process highlights the importance of trigonometric identities, algebraic manipulation, and careful consideration of the domain and periodicity of trigonometric functions.

Practical Applications and Further Exploration

Understanding and solving trigonometric equations like cos(2x - 10°) ⋅ sec(x + 30°) = 1 is not just an academic exercise; it has numerous practical applications in various fields. Trigonometry is a fundamental tool in physics, engineering, navigation, and many other scientific disciplines. For instance, in physics, trigonometric functions are used to describe oscillatory motion, such as the movement of a pendulum or the propagation of waves. In engineering, they are essential for analyzing structures, designing circuits, and processing signals. Navigation relies heavily on trigonometric principles for determining positions and directions.

The ability to solve trigonometric equations is crucial in these applications. For example, consider a scenario in electrical engineering where you need to determine the phase difference between two alternating current (AC) signals. The phase difference can be found by solving a trigonometric equation involving sine or cosine functions. Similarly, in mechanical engineering, calculating the forces acting on a structure often involves solving trigonometric equations to resolve force vectors into their components.

Beyond practical applications, exploring trigonometric equations can lead to a deeper understanding of mathematical concepts. Trigonometric equations are closely related to trigonometric identities, which are fundamental relationships between trigonometric functions. Mastering these identities is essential for simplifying equations and solving problems. Additionally, understanding the periodicity and symmetry of trigonometric functions provides valuable insights into the behavior of these functions and their graphs.

For further exploration, consider investigating more complex trigonometric equations involving multiple trigonometric functions or higher-order terms. Techniques such as substitution, factoring, and using double-angle or half-angle formulas can be employed to solve these equations. Additionally, exploring the graphical solutions of trigonometric equations can provide a visual understanding of the solutions and their behavior. Software tools and graphing calculators can be used to plot trigonometric functions and find their intersections, which represent the solutions to the equations. Furthermore, studying the applications of trigonometric equations in different fields can provide a broader perspective on their significance and utility. This might involve delving into topics such as Fourier analysis, which uses trigonometric functions to decompose complex signals into simpler components, or exploring the role of trigonometry in computer graphics and animation.

Conclusion

In conclusion, solving the trigonometric equation cos(2x - 10°) ⋅ sec(x + 30°) = 1 demonstrates the power and elegance of trigonometric principles. By systematically applying trigonometric identities, algebraic techniques, and a careful consideration of the functions' domains, we were able to find the general solutions for x. This process not only provides a specific answer but also reinforces a broader understanding of trigonometric relationships and problem-solving strategies.

We began by transforming the equation using the reciprocal identity, converting the secant function into its cosine equivalent. This simplification allowed us to rewrite the equation in a more manageable form, where we could directly compare the cosine values of two different angles. By recognizing the conditions under which two cosine values are equal, we derived two sets of possible solutions. These conditions stemmed from the periodic nature and symmetry of the cosine function, leading to general solutions that encompass all possible values of x.

Throughout the solution process, we emphasized the importance of checking for extraneous solutions. The reciprocal relationship between secant and cosine introduces restrictions on the domain, and we carefully verified that our solutions did not violate these restrictions. This step highlights the need for meticulous attention to detail when working with trigonometric functions.

Furthermore, we discussed the practical applications of solving trigonometric equations in various fields, including physics, engineering, and navigation. These applications underscore the real-world relevance of trigonometry and its importance in scientific and technological advancements. By exploring more complex equations and graphical solutions, one can deepen their understanding of trigonometric concepts and their applications.

Ultimately, the ability to solve trigonometric equations is a valuable skill that enhances mathematical proficiency and problem-solving capabilities. It requires a combination of trigonometric knowledge, algebraic dexterity, and logical reasoning. The techniques and strategies discussed in this article provide a solid foundation for tackling a wide range of trigonometric problems, fostering a deeper appreciation for the beauty and utility of mathematics.