Find The Value Of X That Satisfies The Equation Cos(x) = Sin(π/12).

In the realm of mathematics, particularly trigonometry, we often encounter equations that require us to find specific values that satisfy certain conditions. This article delves into a problem where we aim to find the value of x that satisfies the trigonometric equation cos(x) = sin(π/12). This exploration will not only enhance our understanding of trigonometric identities but also showcase how to manipulate these identities to solve equations effectively. Before we jump into the solution, let's understand the core concepts. Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental in describing relationships between angles and sides of triangles. These functions also play a crucial role in modeling periodic phenomena in various fields like physics and engineering. The equation cos(x) = sin(π/12) presents a classic problem where we need to leverage the complementary relationship between sine and cosine functions. This relationship states that the sine of an angle is equal to the cosine of its complementary angle, and vice versa. In mathematical terms, sin(θ) = cos(π/2 - θ) and cos(θ) = sin(π/2 - θ). Understanding this relationship is key to solving our problem. The value π/12 represents an angle in radians, which is a common unit of angular measure in mathematics. To put this angle in perspective, π radians is equivalent to 180 degrees. Therefore, π/12 radians is equivalent to 15 degrees. This conversion helps us visualize the angle and understand its position in the unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane. The unit circle provides a visual representation of trigonometric functions, where the coordinates of a point on the circle correspond to the cosine and sine of the angle formed by the point, the origin, and the positive x-axis. Now, with a firm grasp of these fundamental concepts, we are well-equipped to tackle the equation cos(x) = sin(π/12) and find the value(s) of x that satisfy it. We will begin by utilizing the complementary angle identity to transform the equation into a more manageable form, and then employ algebraic techniques to isolate x and determine its value. This journey through trigonometric problem-solving will not only provide us with the solution to this specific equation but also equip us with the skills to tackle similar problems in the future. This understanding is crucial for various applications in science, engineering, and other fields that rely on trigonometric principles. So, let's embark on this mathematical adventure and unravel the solution to the equation cos(x) = sin(π/12).

Utilizing the Complementary Angle Identity

To solve the equation cos(x) = sin(π/12), the first crucial step involves leveraging the complementary angle identity. As mentioned earlier, this identity states that the sine of an angle is equal to the cosine of its complement, and vice versa. Mathematically, this relationship is expressed as sin(θ) = cos(π/2 - θ) and cos(θ) = sin(π/2 - θ). Applying this identity allows us to transform the equation into a form that is easier to solve. In our case, we can rewrite sin(π/12) using the complementary angle identity. Specifically, we can express sin(π/12) as the cosine of its complement. The complement of an angle is the angle that, when added to the original angle, equals π/2 (or 90 degrees). Therefore, the complement of π/12 is π/2 - π/12. To find this difference, we need to have a common denominator. We can rewrite π/2 as 6π/12. Thus, the complement of π/12 is 6π/12 - π/12 = 5π/12. Now, we can apply the complementary angle identity and rewrite sin(π/12) as cos(5π/12). This transformation is crucial because it allows us to express both sides of the equation in terms of the cosine function. With this substitution, our original equation cos(x) = sin(π/12) now becomes cos(x) = cos(5π/12). This new form of the equation is much easier to solve because it directly relates the cosine of x to the cosine of a specific angle, 5π/12. To solve this equation, we need to understand the properties of the cosine function. The cosine function is periodic, which means it repeats its values at regular intervals. Specifically, the cosine function has a period of 2π, meaning that cos(θ) = cos(θ + 2πk) for any integer k. Additionally, the cosine function is an even function, which means that cos(θ) = cos(-θ). These properties are essential for finding all possible solutions to the equation. Understanding these properties allows us to find all angles whose cosine is equal to cos(5π/12). In the next section, we will explore how to use these properties to determine the general solution for x and identify the specific value that satisfies the given equation. This process involves considering the periodicity and even nature of the cosine function to account for all possible solutions within a given interval. By carefully applying these concepts, we can confidently find the value of x that makes the equation cos(x) = sin(π/12) true.

Solving for x

Now that we have transformed the equation into cos(x) = cos(5π/12), we can proceed to solve for x. As discussed earlier, the cosine function has a period of 2π, meaning that it repeats its values every 2π radians. Additionally, the cosine function is an even function, which means that cos(θ) = cos(-θ). These properties are crucial for finding all possible solutions for x. Given that cos(x) = cos(5π/12), we can deduce that x must be an angle that has the same cosine value as 5π/12. Due to the periodicity of the cosine function, any angle of the form 5π/12 + 2πk, where k is an integer, will have the same cosine value as 5π/12. This accounts for all angles that are coterminal with 5π/12. However, we also need to consider the even nature of the cosine function. Since cos(θ) = cos(-θ), the negative of 5π/12, which is -5π/12, will also have the same cosine value. Therefore, any angle of the form -5π/12 + 2πk, where k is an integer, will also be a solution. Combining these two sets of solutions, we can express the general solution for x as: x = ±5π/12 + 2πk, where k is an integer. This general solution represents all possible values of x that satisfy the equation cos(x) = cos(5π/12). However, in many cases, we are interested in finding a specific solution within a certain interval, such as 0 ≤ x < 2π. To find a specific solution within this interval, we can substitute different integer values for k into the general solution and check if the resulting value of x falls within the specified range. For k = 0, we have two possible solutions: x = 5π/12 and x = -5π/12. Since -5π/12 is negative, it is not within the interval 0 ≤ x < 2π. However, 5π/12 is within this interval and is therefore a valid solution. For k = 1, we have: x = 5π/12 + 2π = 29π/12 and x = -5π/12 + 2π = 19π/12. Both of these solutions are greater than 2π, so they are not within the interval 0 ≤ x < 2π. For k = -1, we have: x = 5π/12 - 2π = -19π/12 and x = -5π/12 - 2π = -29π/12. Both of these solutions are negative and therefore not within the interval 0 ≤ x < 2π. Therefore, within the interval 0 ≤ x < 2π, the solutions are x = 5π/12 and x = 19π/12. However, depending on the context of the problem, we may be looking for a single specific solution. In this case, x = 5π/12 is the most straightforward solution. This solution demonstrates how understanding the properties of trigonometric functions, such as periodicity and evenness, allows us to find all possible solutions to trigonometric equations and identify specific solutions within a given interval.

Conclusion: The Value of x

In conclusion, through a step-by-step approach, we successfully found a value of x that satisfies the equation cos(x) = sin(π/12). We began by understanding the fundamental concepts of trigonometry, including trigonometric functions, the unit circle, and the complementary angle identity. The complementary angle identity, which states that sin(θ) = cos(π/2 - θ), was crucial in transforming the original equation into a more manageable form. By applying this identity, we rewrote sin(π/12) as cos(5π/12), transforming the equation into cos(x) = cos(5π/12). This transformation allowed us to directly relate the cosine of x to the cosine of a specific angle. To solve this equation, we leveraged the properties of the cosine function, including its periodicity and evenness. The periodicity of the cosine function, with a period of 2π, means that the cosine function repeats its values every 2π radians. The evenness of the cosine function means that cos(θ) = cos(-θ). These properties allowed us to determine the general solution for x, which is x = ±5π/12 + 2πk, where k is an integer. This general solution represents all possible values of x that satisfy the equation. To find a specific solution within the interval 0 ≤ x < 2π, we substituted different integer values for k into the general solution and checked if the resulting value of x fell within the specified range. This process led us to the solutions x = 5π/12 and x = 19π/12 within the interval 0 ≤ x < 2π. However, if we are looking for a single specific solution, x = 5π/12 is the most straightforward answer. Therefore, a value of x that satisfies the equation cos(x) = sin(π/12) is 5π/12. This solution highlights the importance of understanding trigonometric identities and properties in solving trigonometric equations. The ability to manipulate these identities and apply these properties is essential for success in trigonometry and related fields. Furthermore, this problem-solving process demonstrates the power of a systematic approach in mathematics. By breaking down the problem into smaller, more manageable steps, we were able to effectively utilize our knowledge and arrive at the solution. This approach can be applied to a wide range of mathematical problems, making it a valuable skill for students and professionals alike. In conclusion, the solution x = 5π/12 not only answers the specific question posed but also reinforces the importance of understanding and applying trigonometric principles. This understanding will serve as a strong foundation for further exploration in mathematics and its applications in various fields.