Solving 2 Sec(x) + 5 = 9 Find Equivalent Equations And Solutions

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Introduction

In this article, we will delve into solving the trigonometric equation 2 sec(x) + 5 = 9. Trigonometric equations are fundamental in mathematics, physics, and engineering, often appearing in various applications such as wave mechanics, electrical circuits, and oscillations. Understanding how to solve these equations is crucial for anyone working in these fields. Our main goal is to find which equation is equivalent to the given equation and then determine the solutions for x. We will systematically break down the problem, explaining each step in detail to ensure clarity and understanding. This involves manipulating the original equation to isolate the trigonometric function, identifying the related angles, and finding the general solutions. The journey through this problem will not only help you solve this specific equation but also equip you with the skills to tackle a variety of similar trigonometric problems. Let's begin by rewriting the given equation and isolating the secant function.

Step-by-Step Solution: Rewriting the Equation

First, let's revisit the given equation: 2 sec(x) + 5 = 9. Our initial task is to isolate the term involving the trigonometric function, which in this case is sec(x). To achieve this, we will perform a series of algebraic manipulations. Start by subtracting 5 from both sides of the equation:

2 sec(x) + 5 - 5 = 9 - 5

This simplifies to:

2 sec(x) = 4

Next, we need to isolate sec(x) completely. To do this, we divide both sides of the equation by 2:

(2 sec(x)) / 2 = 4 / 2

Which simplifies to:

sec(x) = 2

Now that we have isolated sec(x), we can use the reciprocal identity to rewrite the equation in terms of cosine. Recall that sec(x) is the reciprocal of cos(x), meaning sec(x) = 1 / cos(x). Therefore, we can rewrite our equation as:

1 / cos(x) = 2

To solve for cos(x), we can take the reciprocal of both sides of the equation. This gives us:

cos(x) = 1 / 2

This is a crucial step as it transforms the original equation into a more familiar form involving cosine, which is easier to work with. Now we have successfully rewritten the given equation in terms of cosine, and we can identify which of the provided options is equivalent to the original equation. This simplified form will also help us in finding the solutions for x in the subsequent steps. The transformation highlights the importance of understanding trigonometric identities and their application in solving equations. In the next section, we will determine the solutions for x using this equivalent equation.

Identifying the Equivalent Equation

Now that we have simplified the original equation 2 sec(x) + 5 = 9 to cos(x) = 1/2, we can easily identify which of the provided options is equivalent. Let's review the given options:

  1. cos(x) = 1/2
  2. cos(x) = -2
  3. sin(x) = 1/2
  4. sin(x) = -2

By direct comparison, it is clear that the first option, cos(x) = 1/2, matches our simplified equation. Therefore, this is the equivalent equation to the given one. The other options involve different trigonometric functions or values that do not align with our simplification. For instance, cos(x) = -2 is not a valid solution because the range of the cosine function is [-1, 1], meaning the cosine of any angle cannot be -2. Similarly, the sine options do not follow from our simplification steps. This exercise underscores the importance of correctly applying trigonometric identities and algebraic manipulations to arrive at the correct equivalent form. Identifying the equivalent equation is a key step towards solving for the unknown variable, as it sets the stage for finding the specific values of x that satisfy the original equation. In the following sections, we will proceed to find the solutions for x based on the equivalent equation we have identified.

Finding the Solutions for x

Having established that the equivalent equation is cos(x) = 1/2, our next step is to find the solutions for x. To do this, we need to consider the unit circle and the angles at which the cosine function equals 1/2. Recall that cosine corresponds to the x-coordinate on the unit circle. The angles for which the x-coordinate is 1/2 are π/3 (60 degrees) and 5π/3 (300 degrees) within the interval [0, 2π]. These are the principal values for x that satisfy the equation cos(x) = 1/2.

However, we must also consider that trigonometric functions are periodic, meaning they repeat their values at regular intervals. The cosine function has a period of 2Ï€, so we can add integer multiples of 2Ï€ to our principal solutions to find all possible solutions. Therefore, the general solutions for x can be expressed as:

  • x = Ï€/3 + 2Ï€k
  • x = 5Ï€/3 + 2Ï€k

where k is an integer. These general solutions represent all possible angles x for which cos(x) equals 1/2. By substituting different integer values for k, we can find an infinite number of solutions. For example, when k = 0, we get the principal solutions π/3 and 5π/3. When k = 1, we get π/3 + 2π and 5π/3 + 2π, and so on. This periodicity is a fundamental characteristic of trigonometric functions and is crucial to understanding and solving trigonometric equations. The ability to find both principal and general solutions ensures a comprehensive understanding of the equation's behavior. In the next section, we will summarize our findings and reiterate the key steps taken to solve the given trigonometric equation.

Summary of the Solution Process

In this comprehensive guide, we have successfully solved the trigonometric equation 2 sec(x) + 5 = 9. Let's recap the step-by-step process we followed:

  1. Rewrite the equation: We started by isolating the secant function. We subtracted 5 from both sides of the original equation to get 2 sec(x) = 4. Then, we divided both sides by 2 to obtain sec(x) = 2. This initial algebraic manipulation was crucial to simplifying the equation and focusing on the trigonometric component.

  2. Use reciprocal identity: We used the reciprocal identity to convert sec(x) to cos(x). Since sec(x) = 1 / cos(x), we rewrote the equation as 1 / cos(x) = 2. Taking the reciprocal of both sides, we found the equivalent equation cos(x) = 1/2. This step is a key illustration of how trigonometric identities can simplify equations and make them easier to solve.

  3. Identify the equivalent equation: We compared our simplified equation cos(x) = 1/2 with the given options and correctly identified it as the equivalent equation. This ensured that we were working with the correct form for solving the equation.

  4. Find the solutions for x: We determined the principal values of x for which cos(x) = 1/2 by considering the unit circle. The principal solutions are π/3 and 5π/3. Understanding the unit circle and the behavior of trigonometric functions within it is fundamental to finding these solutions.

  5. Express general solutions: We expressed the general solutions for x by adding integer multiples of 2π to the principal solutions, accounting for the periodic nature of the cosine function. The general solutions are x = π/3 + 2πk and x = 5π/3 + 2πk, where k is an integer. This step demonstrates an understanding of the periodic properties of trigonometric functions and how to represent all possible solutions.

By following these steps, we not only solved the given trigonometric equation but also reinforced key concepts in trigonometry and algebraic manipulation. This systematic approach is applicable to a wide range of trigonometric problems and equips you with the tools to tackle similar challenges. The next time you encounter a trigonometric equation, remember this structured process to break it down and find the solutions.

Conclusion

In conclusion, solving the trigonometric equation 2 sec(x) + 5 = 9 involved a systematic approach that included rewriting the equation, using trigonometric identities, identifying the equivalent equation, finding principal solutions, and expressing general solutions. We successfully transformed the original equation into a simpler form, cos(x) = 1/2, which allowed us to determine the values of x that satisfy the equation. The general solutions x = π/3 + 2πk and x = 5π/3 + 2πk (where k is an integer) represent all possible solutions, accounting for the periodic nature of the cosine function. This exercise highlights the importance of understanding trigonometric identities, algebraic manipulation, and the properties of trigonometric functions. These skills are essential for solving a wide variety of problems in mathematics, physics, and engineering. By mastering these techniques, you can confidently tackle complex trigonometric equations and apply them to real-world scenarios. Remember to always consider the periodic nature of trigonometric functions when finding solutions, and utilize the unit circle as a valuable tool for visualizing angles and their corresponding trigonometric values. With practice and a solid understanding of the fundamentals, you can become proficient in solving trigonometric equations and unlock their applications in various fields. This comprehensive guide has hopefully provided you with the knowledge and confidence to approach such problems effectively.