Finding Cos X - Sin X Given Tan X = 1/√3
#Introduction In the realm of trigonometry, unraveling the relationships between trigonometric functions is a fundamental skill. This article delves into a specific problem where we are given the value of the tangent function, $ an x = rac{1}{\sqrt{3}}$, within the interval $0^{\circ} \leq x \leq 90^{\circ}$, and are tasked with finding the value of $\cos x - \sin x$. We will embark on a step-by-step journey, employing trigonometric identities and reasoning, to arrive at the solution. Understanding these relationships is crucial for various applications in mathematics, physics, and engineering. This problem not only tests our knowledge of basic trigonometric values but also our ability to apply them in conjunction with other trigonometric concepts. Let's dive deep into the intricacies of this problem and unlock the solution together.
Deciphering the Given Information: $ an x = rac{1}{\sqrt{3}}$
The cornerstone of our solution lies in the given information: $ an x = rac{1}{\sqrt{3}}$. Tangent, as a trigonometric function, represents the ratio of the sine to the cosine of an angle, or equivalently, the ratio of the opposite side to the adjacent side in a right-angled triangle. We are also given the constraint $0^{\circ} \leq x \leq 90^{\circ}$, which confines our angle $x$ to the first quadrant. This is a crucial piece of information, as it tells us that both sine and cosine of $x$ will be positive in this interval. The given value of $ an x$ is a special value that corresponds to a well-known angle. Recognizing this special value is key to simplifying the problem. Recall that the tangent function is positive in the first quadrant, and the value $ rac{1}{\sqrt{3}}$ is associated with an angle whose sine and cosine have a particular relationship. By understanding the fundamental trigonometric ratios and the constraints on the angle, we can start to unravel the solution to this problem.
Identifying the Angle x: A Critical Step
To effectively solve this problem, we need to identify the angle $x$ that satisfies $\tan x = rac{1}{\sqrt{3}}$. Recalling the standard trigonometric values for special angles, we know that $\tan 30^{\circ} = \frac{1}{\sqrt{3}}$. This is a critical observation, as it directly gives us the value of $x$. Since the problem specifies that $0^{\circ} \leq x \leq 90^{\circ}$, the angle $30^{\circ}$ falls within this range. Therefore, we can confidently say that $x = 30^{\circ}$. Now that we have determined the angle, we can proceed to find the values of $\cos x$ and $\sin x$, which are essential for calculating the final expression $\cos x - \sin x$. Identifying the angle is often the most crucial step in trigonometric problems, as it unlocks the door to applying other trigonometric relationships and identities. The ability to quickly recognize these special angles and their corresponding trigonometric values is a valuable skill in mathematics and its applications.
Determining $\cos x$ and $\sin x$: Essential Trigonometric Values
Now that we have established that $x = 30^{\circ}$, we need to find the values of $\cos 30^{\circ}$ and $\sin 30^{\circ}$. These are standard trigonometric values that are often memorized or can be derived from the unit circle or special right triangles (30-60-90 triangle). We know that $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ and $\sin 30^{\circ} = \frac{1}{2}$. These values are fundamental and widely used in various mathematical contexts. Understanding how these values are derived, whether from the unit circle or special triangles, is crucial for a deeper understanding of trigonometry. With these values in hand, we are now equipped to calculate the final expression, $\cos x - \sin x$. This step highlights the importance of knowing the basic trigonometric values for common angles, as they are building blocks for solving more complex problems. The ability to quickly recall these values saves time and allows for a smoother problem-solving process.
Calculating $\cos x - \sin x$: The Final Step
With the values of $\cos x$ and $\sin x$ determined, we can now calculate the final expression, $\cos x - \sin x$. We have $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ and $\sin 30^{\circ} = \frac{1}{2}$. Substituting these values into the expression, we get:
To simplify this, we can combine the fractions since they have a common denominator:
This is the final simplified answer. This step demonstrates the importance of basic algebraic manipulation in conjunction with trigonometric knowledge. The ability to combine fractions and simplify expressions is a crucial skill in mathematics. By carefully substituting the values and simplifying, we have successfully arrived at the solution to the problem. This final calculation underscores the importance of attention to detail and accuracy in mathematical problem-solving.
In conclusion, by systematically analyzing the given information, identifying the angle $x$, determining the values of $\cos x$ and $\sin x$, and performing the final calculation, we have successfully found the value of $\cos x - \sin x$ when $\tan x = \frac{1}{\sqrt{3}}$ and $0^{\circ} \leq x \leq 90^{\circ}$. The solution is $\frac{\sqrt{3} - 1}{2}$. This problem serves as a valuable exercise in applying trigonometric principles and highlights the importance of understanding the relationships between trigonometric functions, recognizing special angles, and utilizing basic algebraic techniques. The journey through this problem reinforces the interconnectedness of different mathematical concepts and the power of a step-by-step approach to problem-solving. This methodical approach can be applied to a wide range of trigonometric problems and beyond, fostering a deeper understanding and appreciation for the beauty and elegance of mathematics. Mastering these fundamental trigonometric concepts and problem-solving strategies is essential for success in more advanced mathematical studies and applications.
