What Is The Exact Value Of \cot \left(\frac{3 \pi}{4}\right ]?A. − 1 -1 − 1 B. − 2 -\sqrt{2} − 2 ​ C. 1 1 1 D. 2 \sqrt{2} 2 ​

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Introduction

In trigonometry, the cotangent function is defined as the reciprocal of the tangent function. It is denoted by $\cot x$ and is equal to $\frac{\cos x}{\sin x}$. The cotangent function is periodic with a period of $\pi$, and its range is all real numbers. In this article, we will explore the exact value of $\cot \left(\frac{3 \pi}{4}\right)$.

Understanding the Cotangent Function

The cotangent function is a fundamental concept in trigonometry, and it is used to describe the relationship between the adjacent side and the opposite side of a right triangle. The cotangent function is defined as the ratio of the adjacent side to the opposite side. It is denoted by $\cot x$ and is equal to $\frac{\cos x}{\sin x}$.

Properties of the Cotangent Function

The cotangent function has several important properties that make it useful in trigonometry. Some of the key properties of the cotangent function include:

• Periodicity: The cotangent function is periodic with a period of $\pi$. This means that the value of the cotangent function repeats every $\pi$ radians.
• Range: The range of the cotangent function is all real numbers. This means that the cotangent function can take on any real value.
• Reciprocal: The cotangent function is the reciprocal of the tangent function. This means that the cotangent function is equal to the reciprocal of the tangent function.

Finding the Exact Value of $\cot \left(\frac{3 \pi}{4}\right)$

To find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$, we need to use the definition of the cotangent function. The cotangent function is defined as the ratio of the adjacent side to the opposite side. In this case, we are given the angle $\frac{3 \pi}{4}$, and we need to find the ratio of the adjacent side to the opposite side.

Using the Unit Circle

One way to find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$ is to use the unit circle. The unit circle is a circle with a radius of 1 that is centered at the origin. The unit circle is used to define the trigonometric functions, and it is a useful tool for finding exact values of trigonometric functions.

Finding the Coordinates of the Point on the Unit Circle

To find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$, we need to find the coordinates of the point on the unit circle that corresponds to the angle $\frac{3 \pi}{4}$. The coordinates of the point on the unit circle are given by $(\cos \theta, \sin \theta)$, where $\theta$ is the angle.

Using the Definition of the Cotangent Function

Once we have found the coordinates of the point on the unit circle, we can use the definition of the cotangent function to find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$. The cotangent function is defined as the ratio of the adjacent side to the opposite side. In this case, the adjacent side is the x-coordinate of the point on the unit circle, and the opposite side is the y-coordinate of the point on the unit circle.

Finding the Exact Value of $\cot \left(\frac{3 \pi}{4}\right)$

Using the definition of the cotangent function, we can find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$ as follows:

$$\cot \left(\frac{3 \pi}{4}\right) = \frac{\cos \left(\frac{3 \pi}{4}\right)}{\sin \left(\frac{3 \pi}{4}\right)}$$

Evaluating the Trigonometric Functions

To evaluate the trigonometric functions, we need to use the unit circle. The unit circle is a circle with a radius of 1 that is centered at the origin. The unit circle is used to define the trigonometric functions, and it is a useful tool for finding exact values of trigonometric functions.

Finding the Coordinates of the Point on the Unit Circle

To find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$, we need to find the coordinates of the point on the unit circle that corresponds to the angle $\frac{3 \pi}{4}$. The coordinates of the point on the unit circle are given by $(\cos \theta, \sin \theta)$, where $\theta$ is the angle.

Evaluating the Trigonometric Functions

Using the unit circle, we can evaluate the trigonometric functions as follows:

$$\cos \left(\frac{3 \pi}{4}\right) = -\frac{1}{\sqrt{2}}$$

$$\sin \left(\frac{3 \pi}{4}\right) = \frac{1}{\sqrt{2}}$$

Finding the Exact Value of $\cot \left(\frac{3 \pi}{4}\right)$

Using the definition of the cotangent function, we can find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$ as follows:

$$\cot \left(\frac{3 \pi}{4}\right) = \frac{\cos \left(\frac{3 \pi}{4}\right)}{\sin \left(\frac{3 \pi}{4}\right)} = \frac{-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = -1$$

Conclusion

In this article, we have explored the exact value of $\cot \left(\frac{3 \pi}{4}\right)$. We have used the definition of the cotangent function and the unit circle to find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$. The exact value of $\cot \left(\frac{3 \pi}{4}\right)$ is $-1$.

Final Answer

The final answer is $\boxed{-1}$.

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Introduction

In our previous article, we explored the exact value of $\cot \left(\frac{3 \pi}{4}\right)$. We used the definition of the cotangent function and the unit circle to find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$. In this article, we will answer some frequently asked questions about the exact value of $\cot \left(\frac{3 \pi}{4}\right)$.

Q1: What is the definition of the cotangent function?

A1: The cotangent function is defined as the reciprocal of the tangent function. It is denoted by $\cot x$ and is equal to $\frac{\cos x}{\sin x}$.

Q2: How do you find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$?

A2: To find the exact value of $\cot \left(\frac{3 \pi}{4}\right)$, you need to use the definition of the cotangent function and the unit circle. The unit circle is a circle with a radius of 1 that is centered at the origin. The unit circle is used to define the trigonometric functions, and it is a useful tool for finding exact values of trigonometric functions.

Q3: What is the unit circle?

A3: The unit circle is a circle with a radius of 1 that is centered at the origin. The unit circle is used to define the trigonometric functions, and it is a useful tool for finding exact values of trigonometric functions.

Q4: How do you evaluate the trigonometric functions using the unit circle?

A4: To evaluate the trigonometric functions using the unit circle, you need to find the coordinates of the point on the unit circle that corresponds to the angle. The coordinates of the point on the unit circle are given by $(\cos \theta, \sin \theta)$, where $\theta$ is the angle.

Q5: What is the exact value of $\cot \left(\frac{3 \pi}{4}\right)$?

A5: The exact value of $\cot \left(\frac{3 \pi}{4}\right)$ is $-1$.

Q6: Why is the exact value of $\cot \left(\frac{3 \pi}{4}\right)$ important?

A6: The exact value of $\cot \left(\frac{3 \pi}{4}\right)$ is important because it is used in many mathematical applications, such as trigonometry and calculus.

Q7: How do you use the exact value of $\cot \left(\frac{3 \pi}{4}\right)$ in mathematical applications?

A7: The exact value of $\cot \left(\frac{3 \pi}{4}\right)$ is used in many mathematical applications, such as trigonometry and calculus. It is used to solve problems and equations that involve trigonometric functions.

Q8: Can you provide an example of how to use the exact value of $\cot \left(\frac{3 \pi}{4}\right)$ in a mathematical application?

A8: Yes, here is an example of how to use the exact value of $\cot \left(\frac{3 \pi}{4}\right)$ in a mathematical application:

Suppose we want to find the value of $\cot \left(\frac{3 \pi}{4}\right) \sin \left(\frac{\pi}{4}\right)$. We can use the exact value of $\cot \left(\frac{3 \pi}{4}\right)$, which is $-1$, and the value of $\sin \left(\frac{\pi}{4}\right)$, which is $\frac{1}{\sqrt{2}}$, to find the value of the expression.

$$\cot \left(\frac{3 \pi}{4}\right) \sin \left(\frac{\pi}{4}\right) = (-1) \left(\frac{1}{\sqrt{2}}\right) = -\frac{1}{\sqrt{2}}$$

Conclusion

In this article, we have answered some frequently asked questions about the exact value of $\cot \left(\frac{3 \pi}{4}\right)$. We have provided definitions, examples, and explanations to help you understand the exact value of $\cot \left(\frac{3 \pi}{4}\right)$ and how to use it in mathematical applications.

Final Answer

The final answer is $\boxed{-1}$.