What Is The Exact Value Of $\tan \left(-\frac{\pi}{3}\right$\]?A. $-\sqrt{3}$ B. $-\frac{\sqrt{3}}{3}$ C. $\frac{\sqrt{3}}{3}$ D. $\sqrt{3}$

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental concepts in trigonometry is the tangent function, which is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle. In this article, we will explore the exact value of $\tan \left(-\frac{\pi}{3}\right)$ and discuss the different options available.

What is the Tangent Function?

The tangent function is a trigonometric function that is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle. It is denoted by the symbol $\tan$ and is defined as:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

where $\theta$ is the angle in question.

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry that is used to define the values of the trigonometric functions. The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. The angle $\theta$ is measured counterclockwise from the positive x-axis to the terminal side of the angle.

Evaluating $\tan \left(-\frac{\pi}{3}\right)$

To evaluate $\tan \left(-\frac{\pi}{3}\right)$, we need to use the unit circle and the definition of the tangent function. We know that the angle $-\frac{\pi}{3}$ is in the second quadrant of the unit circle, where the sine function is positive and the cosine function is negative.

Using the unit circle, we can determine that the sine of $-\frac{\pi}{3}$ is $-\frac{\sqrt{3}}{2}$ and the cosine of $-\frac{\pi}{3}$ is $-\frac{1}{2}$. Therefore, we can evaluate the tangent function as follows:

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

Conclusion

In conclusion, the exact value of $\tan \left(-\frac{\pi}{3}\right)$ is $\sqrt{3}$. This value can be determined using the unit circle and the definition of the tangent function. The tangent function is a fundamental concept in trigonometry that is used to define the relationships between the sides and angles of triangles.

Answer

The correct answer is:

• D. $\sqrt{3}$

Why is this the correct answer?

This is the correct answer because the tangent function is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle. In this case, the angle $-\frac{\pi}{3}$ is in the second quadrant of the unit circle, where the sine function is positive and the cosine function is negative. Therefore, the tangent function is equal to the ratio of the sine function to the cosine function, which is equal to $\sqrt{3}$.

What are the other options?

The other options are:

• A. $-\sqrt{3}$: This is not the correct answer because the tangent function is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle. In this case, the angle $-\frac{\pi}{3}$ is in the second quadrant of the unit circle, where the sine function is positive and the cosine function is negative. Therefore, the tangent function is equal to the ratio of the sine function to the cosine function, which is equal to $\sqrt{3}$.
• B. $-\frac{\sqrt{3}}{3}$: This is not the correct answer because the tangent function is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle. In this case, the angle $-\frac{\pi}{3}$ is in the second quadrant of the unit circle, where the sine function is positive and the cosine function is negative. Therefore, the tangent function is equal to the ratio of the sine function to the cosine function, which is equal to $\sqrt{3}$.
• C. $\frac{\sqrt{3}}{3}$: This is not the correct answer because the tangent function is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle. In this case, the angle $-\frac{\pi}{3}$ is in the second quadrant of the unit circle, where the sine function is positive and the cosine function is negative. Therefore, the tangent function is equal to the ratio of the sine function to the cosine function, which is equal to $\sqrt{3}$.

Why are these options incorrect?

These options are incorrect because they do not take into account the definition of the tangent function and the properties of the unit circle. The tangent function is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle. In this case, the angle $-\frac{\pi}{3}$ is in the second quadrant of the unit circle, where the sine function is positive and the cosine function is negative. Therefore, the tangent function is equal to the ratio of the sine function to the cosine function, which is equal to $\sqrt{3}$.

What is the importance of understanding the tangent function?

Understanding the tangent function is important because it is a fundamental concept in trigonometry that is used to define the relationships between the sides and angles of triangles. The tangent function is used in a variety of applications, including physics, engineering, and computer science. It is also used in the calculation of trigonometric identities and the solution of trigonometric equations.

Conclusion

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Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle. It is denoted by the symbol $\tan$ and is defined as:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Q: What is the unit circle?

A: The unit circle is a fundamental concept in trigonometry that is used to define the values of the trigonometric functions. The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. The angle $\theta$ is measured counterclockwise from the positive x-axis to the terminal side of the angle.

Q: How do I evaluate the tangent function?

A: To evaluate the tangent function, you need to use the unit circle and the definition of the tangent function. You can use the sine and cosine functions to determine the value of the tangent function.

Q: What is the exact value of $\tan \left(-\frac{\pi}{3}\right)$?

A: The exact value of $\tan \left(-\frac{\pi}{3}\right)$ is $\sqrt{3}$. This value can be determined using the unit circle and the definition of the tangent function.

Q: Why is the tangent function important?

A: The tangent function is important because it is a fundamental concept in trigonometry that is used to define the relationships between the sides and angles of triangles. The tangent function is used in a variety of applications, including physics, engineering, and computer science.

Q: What are some common mistakes to avoid when working with the tangent function?

A: Some common mistakes to avoid when working with the tangent function include:

• Not using the unit circle to determine the values of the sine and cosine functions.
• Not using the definition of the tangent function to evaluate the tangent function.
• Not taking into account the properties of the unit circle, such as the fact that the sine function is positive in the second quadrant and the cosine function is negative in the second quadrant.

Q: How can I practice working with the tangent function?

A: You can practice working with the tangent function by:

• Using the unit circle to determine the values of the sine and cosine functions.
• Using the definition of the tangent function to evaluate the tangent function.
• Working with different angles and values to become more familiar with the tangent function.

Q: What are some real-world applications of the tangent function?

A: Some real-world applications of the tangent function include:

• Physics: The tangent function is used to describe the motion of objects in terms of their position, velocity, and acceleration.
• Engineering: The tangent function is used to design and analyze the performance of mechanical systems, such as gears and pulleys.
• Computer Science: The tangent function is used in computer graphics and game development to create realistic and interactive 3D models.

Q: How can I use the tangent function in my own work or studies?

A: You can use the tangent function in your own work or studies by:

• Using the unit circle to determine the values of the sine and cosine functions.
• Using the definition of the tangent function to evaluate the tangent function.
• Working with different angles and values to become more familiar with the tangent function.

Conclusion

In conclusion, the tangent function is a fundamental concept in trigonometry that is used to define the relationships between the sides and angles of triangles. Understanding the tangent function is important because it is used in a variety of applications, including physics, engineering, and computer science. By practicing working with the tangent function and using it in real-world applications, you can become more familiar with this important concept and use it to solve problems and make informed decisions.