Finding Cos X - Sin X Given Tan X = 1/√3

Introduction

In trigonometry, solving for trigonometric functions given certain conditions is a common task. This article delves into finding the value of $\cos x - \sin x$ given that $\tan x = \frac{1}{\sqrt{3}}$ and $0^\circ \leq x \leq 90^\circ$. This problem involves using trigonometric identities and understanding the properties of trigonometric functions within the specified interval. We will start by determining the value of $x$ from the given tangent value and then proceed to calculate the values of $\sin x$ and $\cos x$. Finally, we will compute the difference between $\cos x$ and $\sin x$. Let's explore this problem step by step to gain a clearer understanding.

Determining the Value of x

The key to solving this problem lies in first identifying the angle $x$ for which $\tan x = \frac{1}{\sqrt{3}}$. Recall that the tangent function, $\tan x$, is defined as the ratio of the sine function to the cosine function, i.e., $\tan x = \frac{\sin x}{\cos x}$. Given that $\tan x = \frac{1}{\sqrt{3}}$, we are looking for an angle $x$ in the interval $0^\circ \leq x \leq 90^\circ$ where this ratio holds true. This involves recalling the values of trigonometric functions for standard angles such as $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$. By examining these standard angles, we can recognize that $\tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}$. Therefore, the angle $x$ that satisfies the given condition is $30^\circ$. This foundational step is crucial because the values of $\sin x$ and $\cos x$ depend directly on the value of $x$. Once we have correctly identified the angle, we can proceed to find the sine and cosine values for that specific angle, which are necessary to calculate the final expression $\cos x - \sin x$. Understanding the unit circle and the values of trigonometric functions at key angles is immensely helpful in this context.

Calculating sin x and cos x

Having determined that $x = 30^\circ$, our next step is to calculate the values of $\sin x$ and $\cos x$. The sine function, $\sin x$, represents the y-coordinate of a point on the unit circle corresponding to the angle $x$, while the cosine function, $\cos x$, represents the x-coordinate of the same point. For the angle $x = 30^\circ$, we can recall the standard trigonometric values or use the unit circle to find the sine and cosine. Specifically, $\sin 30^\circ = \frac{1}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$. These values are fundamental and are frequently used in trigonometric calculations. It’s important to remember these values, as they are derived from the geometry of special right triangles, such as the 30-60-90 triangle. Knowing these values allows us to proceed with the final calculation of the expression $\cos x - \sin x$. By substituting the values of $\sin 30^\circ$ and $\cos 30^\circ$, we can simplify the expression and arrive at the solution. The accurate determination of $\sin x$ and $\cos x$ is pivotal in arriving at the correct final answer.

Computing cos x - sin x

Now that we have established the values of $\sin x$ and $\cos x$ for $x = 30^\circ$, we can compute the expression $\cos x - \sin x$. We found that $\sin 30^\circ = \frac{1}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$. Substituting these values into the expression, we get:

$$\cos x - \sin x = \frac{\sqrt{3}}{2} - \frac{1}{2}$$

To simplify this, we can combine the terms since they have a common denominator:

$$\cos x - \sin x = \frac{\sqrt{3} - 1}{2}$$

This is the final value of $\cos x - \sin x$ for the given conditions. The result is an exact value, which is often preferred in mathematical contexts over a decimal approximation. The calculation demonstrates how understanding trigonometric values for specific angles and basic algebraic manipulation can lead to the solution. This result is significant as it provides a specific value for the difference between the cosine and sine of the angle $30^\circ$, which is a valuable piece of information in various mathematical and scientific applications. The entire process, from identifying the angle to computing the final expression, illustrates the interconnectedness of different trigonometric concepts.

Conclusion

In this article, we successfully found the value of $\cos x - \sin x$ given that $\tan x = \frac{1}{\sqrt{3}}$ and $0^\circ \leq x \leq 90^\circ$. We began by identifying the angle $x$ as $30^\circ$ using the given tangent value. Subsequently, we calculated the sine and cosine of this angle, finding that $\sin 30^\circ = \frac{1}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$. Finally, we computed the difference $\cos x - \sin x$, which resulted in $\frac{\sqrt{3} - 1}{2}$. This exercise underscores the importance of understanding trigonometric identities, standard angle values, and algebraic manipulation in solving trigonometric problems. The process involved a step-by-step approach, starting from identifying the angle, finding the sine and cosine values, and then computing the required expression. This methodical approach is crucial for accuracy and clarity in mathematical problem-solving. The result we obtained is not only a specific solution but also a demonstration of the broader principles of trigonometry, which are applicable in various fields of science and engineering. The ability to solve such problems enhances one’s mathematical skills and provides a solid foundation for more advanced topics in trigonometry and calculus.