Solving 2 Sec(x) + 5 = 9 Finding The Equivalent Equation

In this comprehensive guide, we will walk you through the step-by-step process of solving the trigonometric equation 2 sec(x) + 5 = 9. Our primary goal is to simplify the equation and identify the equivalent trigonometric equation from the given options. This involves understanding trigonometric identities, algebraic manipulations, and the properties of trigonometric functions. Let's embark on this mathematical journey to demystify the solution and enhance your problem-solving skills. By the end of this article, you'll have a solid grasp of how to tackle similar problems, making you more confident in your mathematical abilities.

Understanding the Initial Equation

Before diving into the solution, let's first understand the initial equation: 2 sec(x) + 5 = 9. This equation involves the secant function, which is the reciprocal of the cosine function. To solve for x, we need to isolate the secant function and then convert it into a cosine function. This transformation is crucial because cosine is a more commonly used trigonometric function, and it makes the equation easier to solve. Moreover, understanding the properties of trigonometric functions and their relationships is key to unlocking solutions in trigonometry. So, let's start by isolating the secant function.

Step 1: Isolate the Secant Function

To isolate the secant function in the equation 2 sec(x) + 5 = 9, we need to perform a series of algebraic manipulations. The first step is to subtract 5 from both sides of the equation. This gives us 2 sec(x) = 9 - 5, which simplifies to 2 sec(x) = 4. Now, to completely isolate the secant function, we divide both sides by 2. This results in sec(x) = 4 / 2, which further simplifies to sec(x) = 2. At this stage, we have successfully isolated the secant function. The next step is to convert this into a cosine function, which will lead us closer to the solution.

Step 2: Convert Secant to Cosine

Since secant is the reciprocal of cosine, we can rewrite sec(x) = 2 in terms of cosine. The relationship between secant and cosine is given by sec(x) = 1 / cos(x). Therefore, if sec(x) = 2, then 1 / cos(x) = 2. To find cos(x), we take the reciprocal of both sides of the equation. This gives us cos(x) = 1 / 2. This conversion is a critical step in solving the equation because it allows us to work with a more familiar trigonometric function. Now that we have cos(x) = 1 / 2, we can proceed to identify the equivalent equation from the given options.

Identifying the Equivalent Equation

Now that we have simplified the original equation to cos(x) = 1 / 2, we can compare it with the given options to find the equivalent equation. The options provided are:

• cos(x) = 1/2
• cos(x) = -2
• sin(x) = 1/2
• sin(x) = -2

By direct comparison, it is clear that the equation cos(x) = 1 / 2 matches our result. This means that this is the equivalent equation to the original equation 2 sec(x) + 5 = 9. The other options involve different trigonometric functions or values that do not match our simplified equation. Therefore, the correct equivalent equation is cos(x) = 1 / 2. This simple comparison highlights the importance of simplification in problem-solving.

Why Other Options Are Incorrect

To further solidify our understanding, let's briefly discuss why the other options are incorrect. The option cos(x) = -2 is incorrect because the range of the cosine function is between -1 and 1, inclusive. This means that the value of cos(x) can never be -2. Similarly, the option sin(x) = -2 is also incorrect because the range of the sine function is also between -1 and 1, inclusive. Therefore, sin(x) cannot be -2. The option sin(x) = 1 / 2 is a valid value for the sine function, but it does not follow from our simplification of the original equation. The correct equation we derived was cos(x) = 1 / 2, making this the only equivalent equation among the options.

Importance of Understanding Trigonometric Identities

In solving this equation, we utilized the fundamental trigonometric identity sec(x) = 1 / cos(x). Understanding and applying trigonometric identities is crucial in simplifying and solving trigonometric equations. These identities provide relationships between different trigonometric functions, allowing us to transform equations into more manageable forms. Mastery of these identities enhances your ability to manipulate and solve a wide range of trigonometric problems. Therefore, it is essential to familiarize yourself with various trigonometric identities and their applications.

Common Trigonometric Identities

Here are a few common trigonometric identities that are frequently used in solving equations:

• Pythagorean Identities: sin²(x) + cos²(x) = 1, 1 + tan²(x) = sec²(x), 1 + cot²(x) = csc²(x)
• Reciprocal Identities: csc(x) = 1 / sin(x), sec(x) = 1 / cos(x), cot(x) = 1 / tan(x)
• Quotient Identities: tan(x) = sin(x) / cos(x), cot(x) = cos(x) / sin(x)
• Double Angle Identities: sin(2x) = 2 sin(x) cos(x), cos(2x) = cos²(x) - sin²(x), tan(2x) = 2 tan(x) / (1 - tan²(x))

Knowing these identities and how to apply them can significantly simplify the process of solving trigonometric equations. For example, in our case, using the reciprocal identity sec(x) = 1 / cos(x) was essential to transform the equation into a more solvable form.

General Steps to Solve Trigonometric Equations

Solving trigonometric equations often involves a systematic approach. Here are the general steps you can follow to solve such equations:

1. Simplify the Equation: Begin by simplifying the equation using algebraic manipulations and trigonometric identities. This may involve combining like terms, expanding expressions, or substituting identities to reduce the complexity of the equation.
2. Isolate the Trigonometric Function: Isolate the trigonometric function (e.g., sin(x), cos(x), tan(x)) on one side of the equation. This may involve adding, subtracting, multiplying, or dividing both sides of the equation by appropriate values or expressions.
3. Solve for the Angle: Once the trigonometric function is isolated, solve for the angle using inverse trigonometric functions (e.g., arcsin, arccos, arctan) or by recognizing common trigonometric values. Remember to consider all possible solutions within the given interval or domain.
4. Check for Extraneous Solutions: After finding the solutions, it is crucial to check them in the original equation to ensure they are valid. Sometimes, algebraic manipulations can introduce extraneous solutions, which do not satisfy the original equation.
5. Express the General Solution: If the problem requires a general solution, express the solutions in terms of the periodicity of the trigonometric functions. For example, if cos(x) = 1 / 2, the general solution can be expressed as x = ± π / 3 + 2πk, where k is an integer.

By following these steps, you can systematically approach and solve a wide variety of trigonometric equations. Practice is key to mastering these techniques and becoming proficient in solving trigonometric problems.

Conclusion

In conclusion, we successfully solved the equation 2 sec(x) + 5 = 9 by isolating the secant function, converting it to cosine, and identifying the equivalent equation as cos(x) = 1 / 2. We also discussed why the other options were incorrect and emphasized the importance of understanding trigonometric identities. Furthermore, we outlined the general steps to solve trigonometric equations, providing you with a roadmap for tackling similar problems. By understanding the underlying concepts and practicing these techniques, you can enhance your problem-solving skills and build a solid foundation in trigonometry. Remember, mathematics is a journey of continuous learning and exploration, so keep practicing and challenging yourself with new problems!