Solving Tan(x) = Cot(x + 20°) A Step-by-Step Trigonometric Solution

Trigonometric equations, such as the one we're about to solve, tan(x) = cot(x + 20°), are a cornerstone of mathematics, particularly in trigonometry and calculus. These equations involve trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant, and their solutions often require a solid understanding of trigonometric identities and relationships. Solving trigonometric equations isn't just an academic exercise; it has practical applications in various fields, including physics, engineering, and computer graphics, where periodic phenomena are modeled using trigonometric functions. In this comprehensive guide, we'll break down the steps to solve the given equation, emphasizing the underlying principles and techniques. Our primary goal is to provide a clear, step-by-step solution that is accessible to students and enthusiasts alike. We will explore the properties of trigonometric functions, apply relevant identities, and arrive at the general solutions for x. Moreover, we'll delve into the nuances of expressing these solutions in degrees and radians, ensuring a thorough understanding of the problem. This article aims to not only solve the specific equation but also equip you with the tools to tackle a wide range of trigonometric challenges. By the end of this guide, you'll have a solid grasp of the methods involved and be well-prepared to approach similar problems with confidence. Let's embark on this mathematical journey together, unraveling the intricacies of trigonometric equations and discovering the beauty of their solutions.

Understanding the Basics: Tangent and Cotangent Relationship

To effectively solve the equation tan(x) = cot(x + 20°), we first need to understand the fundamental relationship between the tangent (tan) and cotangent (cot) functions. Tangent and cotangent are reciprocal functions, meaning that cot(θ) = 1/tan(θ) and tan(θ) = 1/cot(θ) for any angle θ where these functions are defined. This reciprocal relationship is crucial because it allows us to express the given equation in terms of a single trigonometric function, making it easier to solve. The cotangent function can also be expressed in terms of the tangent function using the identity cot(θ) = tan(90° - θ). This identity stems from the complementary angle relationship in trigonometry, where the cotangent of an angle is equal to the tangent of its complement. Applying this identity is a common strategy in solving trigonometric equations, as it helps to unify the trigonometric functions involved. In our specific case, we can rewrite cot(x + 20°) as tan(90° - (x + 20°)), which simplifies to tan(70° - x). This transformation is a key step in the solution process, as it allows us to set up an equation involving only the tangent function. Understanding the periodicity of trigonometric functions is another essential aspect of solving trigonometric equations. The tangent function, for instance, has a period of 180° (or π radians), meaning that tan(θ) = tan(θ + 180°n) for any integer n. This periodicity implies that there are infinitely many solutions to trigonometric equations, and we need to account for all possible solutions within a given range or find the general solution. By leveraging these basic relationships and identities, we can simplify the given equation and pave the way for finding its solutions. In the subsequent sections, we will delve deeper into the solution process, applying these principles to solve for x.

Step-by-Step Solution: Transforming the Equation

Now that we have a firm grasp of the fundamental trigonometric relationships, let's dive into the step-by-step solution of the equation tan(x) = cot(x + 20°). Our initial goal is to transform the equation so that it involves only one trigonometric function. As discussed earlier, we can use the identity cot(θ) = tan(90° - θ) to rewrite the cotangent term. Applying this identity to our equation, we get: tan(x) = tan(90° - (x + 20°)). Simplifying the expression inside the tangent function on the right side, we have: tan(x) = tan(90° - x - 20°), which further simplifies to: tan(x) = tan(70° - x). This transformation is a crucial step, as it now allows us to compare the arguments of the tangent functions. When two tangent functions are equal, their arguments must either be equal or differ by a multiple of the tangent function's period, which is 180° (or π radians). Therefore, we can set up the following equation: x = 70° - x + 180°n, where n is an integer. This equation represents the general condition for the equality of the tangent functions. By introducing the term 180°n, we account for all possible solutions arising from the periodicity of the tangent function. In the next step, we will solve this equation for x, keeping in mind that we are looking for a general solution that encompasses all possible values of x that satisfy the original equation. The careful application of trigonometric identities and the understanding of periodicity are key to successfully solving this type of equation. By breaking down the problem into manageable steps, we can systematically arrive at the solution.

Solving for x: Finding the General Solution

Having transformed our original equation into x = 70° - x + 180°n, where n is an integer, the next step is to isolate x and find the general solution. To do this, we first add x to both sides of the equation: 2x = 70° + 180°n. Next, we divide both sides by 2 to solve for x: x = 35° + 90°n. This equation represents the general solution for x in degrees. It tells us that the solutions to the original trigonometric equation are angles that are 35° plus any integer multiple of 90°. The general solution is a powerful representation because it encompasses all possible solutions to the equation. For any integer value of n, we get a solution for x that satisfies the original equation tan(x) = cot(x + 20°). For example, if n = 0, we get x = 35°. If n = 1, we get x = 35° + 90° = 125°. If n = -1, we get x = 35° - 90° = -55°, and so on. These are just a few examples, and there are infinitely many solutions as n ranges over all integers. It's important to note that the general solution captures the periodic nature of the trigonometric functions. The tangent and cotangent functions repeat their values every 180°, which is reflected in the 90°n term in our general solution. In the next section, we will discuss how to express this general solution in radians, which is another common way to represent angles in mathematics and physics. Understanding how to convert between degrees and radians is a crucial skill in trigonometry and related fields.

Expressing the Solution in Radians

While the general solution x = 35° + 90°n is perfectly valid in degrees, it's often necessary to express angles in radians, especially in calculus and more advanced mathematical contexts. Radians provide a natural way to measure angles, relating the angle to the arc length along a circle. To convert from degrees to radians, we use the conversion factor π radians = 180°. Therefore, 1 degree is equal to π/180 radians. Applying this conversion to our general solution, we first convert the constant term 35° to radians: 35° * (π/180) = 7π/36 radians. Next, we convert the term 90° to radians: 90° * (π/180) = π/2 radians. Substituting these radian values into our general solution, we get: x = 7π/36 + (π/2)n, where n is an integer. This is the general solution for x expressed in radians. It represents the same set of solutions as the degree version, but in a different unit of measure. Just as with the degree solution, we can plug in different integer values for n to find specific solutions in radians. For example, if n = 0, we get x = 7π/36 radians. If n = 1, we get x = 7π/36 + π/2 = 25π/36 radians. If n = -1, we get x = 7π/36 - π/2 = -11π/36 radians, and so on. These solutions correspond to the degree solutions we found earlier, but they are expressed in radians. Understanding how to work with radians is essential for many applications of trigonometry, including calculus, physics, and engineering. In these fields, radians often simplify formulas and calculations, making them the preferred unit for angular measure. By mastering the conversion between degrees and radians, you'll be well-equipped to tackle a wide range of problems involving trigonometric functions.

Verification and Conclusion: Ensuring the Solution's Validity

After finding the general solution to a trigonometric equation, it's always a good practice to verify the solution to ensure its validity. This involves plugging the general solution back into the original equation and checking if it holds true. Our original equation was tan(x) = cot(x + 20°), and our general solution is x = 35° + 90°n, where n is an integer. To verify, we substitute the general solution into the original equation: tan(35° + 90°n) = cot((35° + 90°n) + 20°). Simplifying the argument of the cotangent function, we get: tan(35° + 90°n) = cot(55° + 90°n). Now, we can use the identity cot(θ) = tan(90° - θ) to rewrite the cotangent term: tan(35° + 90°n) = tan(90° - (55° + 90°n)). This simplifies to: tan(35° + 90°n) = tan(35° - 90°n). Since the tangent function has a period of 180°, we know that tan(θ) = tan(θ + 180°k) for any integer k. Therefore, we need to check if 35° + 90°n = 35° - 90°n + 180°k for some integer k. Simplifying this equation, we get: 180°n = 180°k, which implies that n = k. This means that the equation holds true for any integer value of n, confirming the validity of our general solution. In conclusion, we have successfully solved the trigonometric equation tan(x) = cot(x + 20°). By applying trigonometric identities, understanding the periodicity of trigonometric functions, and carefully solving the resulting equation, we arrived at the general solution x = 35° + 90°n in degrees or x = 7π/36 + (π/2)n in radians, where n is an integer. This solution encompasses all possible values of x that satisfy the original equation. The process of solving trigonometric equations requires a solid foundation in trigonometric principles and a systematic approach. By mastering these techniques, you'll be well-equipped to tackle a wide range of trigonometric problems.