Not using the correct period

Oct 13, 2025 by windailys.com 119 views

Introduction

Trigonometric functions are essential in mathematics, and understanding their behavior is crucial for solving various problems. The tangent function, in particular, is a fundamental trigonometric function that is used to describe the ratio of the opposite side to the adjacent side in a right-angled triangle. In this article, we will evaluate the expression $\tan \left(-\frac{7 \pi}{3}\right)$ and explore its properties.

Understanding the Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions:

\tan(x) = \frac{\sin(x)}{\cos(x)}

This function is periodic with a period of $\pi$ , meaning that the value of the tangent function repeats every $\pi$ radians. The tangent function is also an odd function, which means that $\tan(-x) = -\tan(x)$ .

Evaluating the Expression

To evaluate the expression $\tan \left(-\frac{7 \pi}{3}\right)$ , we need to first convert the angle from radians to degrees. We can do this using the conversion factor:

1 \text{ radian} = \frac{180}{\pi} \text{ degrees}

Using this conversion factor, we can convert the angle to degrees:

-\frac{7 \pi}{3} = -\frac{7 \pi}{3} \times \frac{180}{\pi} = -420 \text{ degrees}

Since the tangent function is periodic with a period of $\pi$ , we can add or subtract multiples of $\pi$ to the angle without changing the value of the tangent function. Therefore, we can rewrite the angle as:

-\frac{7 \pi}{3} = -\frac{7 \pi}{3} + 2 \pi = -\frac{7 \pi}{3} + \frac{6 \pi}{3} = -\frac{\pi}{3}

Now, we can evaluate the tangent function at the angle $-\frac{\pi}{3}$ :

\tan \left(-\frac{7 \pi}{3}\right) = \tan \left(-\frac{\pi}{3}\right)

Using the properties of the tangent function, we know that $\tan(-x) = -\tan(x)$ . Therefore, we can rewrite the expression as:

\tan \left(-\frac{7 \pi}{3}\right) = -\tan \left(\frac{\pi}{3}\right)

Evaluating the Tangent Function at $\frac{\pi}{3}$

To evaluate the tangent function at $\frac{\pi}{3}$ , we need to use the definition of the tangent function:

\tan(x) = \frac{\sin(x)}{\cos(x)}

Using the values of the sine and cosine functions at $\frac{\pi}{3}$ , we get:

\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}} = \sqrt{3}

Conclusion

In conclusion, we have evaluated the expression $\tan \left(-\frac{7 \pi}{3}\right)$ and found that it is equal to $-\sqrt{3}$ . This result is consistent with the properties of the tangent function, which is an odd function that repeats every $\pi$ radians.

Properties of the Tangent Function

The tangent function has several important properties that are useful for solving problems. Some of these properties include:

Periodicity: The tangent function is periodic with a period of $\pi$ , meaning that the value of the tangent function repeats every $\pi$ radians.
Odd function: The tangent function is an odd function, which means that $\tan(-x) = -\tan(x)$ .
Domain: The domain of the tangent function is all real numbers except for odd multiples of $\frac{\pi}{2}$ .
Range: The range of the tangent function is all real numbers.

Applications of the Tangent Function

The tangent function has several important applications in mathematics and science. Some of these applications include:

Trigonometry: The tangent function is used to describe the ratio of the opposite side to the adjacent side in a right-angled triangle.
Calculus: The tangent function is used to define the derivative of a function.
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 various systems, such as bridges, buildings, and electronic circuits.

Final Thoughts

In conclusion, the tangent function is a fundamental trigonometric function that is used to describe the ratio of the opposite side to the adjacent side in a right-angled triangle. The expression $\tan \left(-\frac{7 \pi}{3}\right)$ is equal to $-\sqrt{3}$ , which is consistent with the properties of the tangent function. The tangent function has several important properties and applications, and it is an essential tool for solving problems in mathematics and science.

Introduction

In our previous article, we evaluated the expression $\tan \left(-\frac{7 \pi}{3}\right)$ and found that it is equal to $-\sqrt{3}$ . In this article, we will answer some common questions related to this expression and provide additional insights into the properties and applications of the tangent function.

Q1: What is the period of the tangent function?

A1: The period of the tangent function is $\pi$ , meaning that the value of the tangent function repeats every $\pi$ radians.

Q2: Is the tangent function an odd function?

A2: Yes, the tangent function is an odd function, which means that $\tan(-x) = -\tan(x)$ .

Q3: What is the domain of the tangent function?

A3: The domain of the tangent function is all real numbers except for odd multiples of $\frac{\pi}{2}$ .

Q4: What is the range of the tangent function?

A4: The range of the tangent function is all real numbers.

Q5: How do you evaluate the tangent function at a negative angle?

A5: To evaluate the tangent function at a negative angle, you can use the property $\tan(-x) = -\tan(x)$ .

Q6: What is the relationship between the tangent function and the sine and cosine functions?

A6: The tangent function is defined as the ratio of the sine and cosine functions:

\tan(x) = \frac{\sin(x)}{\cos(x)}

Q7: How do you use the tangent function in real-world applications?

A7: The tangent function is used in various real-world applications, such as:

Trigonometry: The tangent function is used to describe the ratio of the opposite side to the adjacent side in a right-angled triangle.
Calculus: The tangent function is used to define the derivative of a function.
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 various systems, such as bridges, buildings, and electronic circuits.

Q8: Can you provide examples of how to use the tangent function in different contexts?

A8: Here are some examples of how to use the tangent function in different contexts:

Trigonometry: To find the length of the opposite side of a right-angled triangle, you can use the tangent function:

\tan(x) = \frac{\text{opposite}}{\text{adjacent}}

Calculus: To find the derivative of a function, you can use the tangent function:

f'(x) = \tan(f(x))

Physics: To describe the motion of an object, you can use the tangent function:

\tan(\theta) = \frac{\text{velocity}}{\text{position}}

Q9: What are some common mistakes to avoid when working with the tangent function?

A9: Here are some common mistakes to avoid when working with the tangent function:

Not using the correct period: Make sure to use the correct period of the tangent function, which is $\pi$ .
Not using the correct domain: Make sure to use the correct domain of the tangent function, which is all real numbers except for odd multiples of $\frac{\pi}{2}$ .
Not using the correct range: Make sure to use the correct range of the tangent function, which is all real numbers.

Q10: Can you provide additional resources for learning more about the tangent function?

A10: Here are some additional resources for learning more about the tangent function:

Textbooks: There are many textbooks available that cover the tangent function in detail.
Online resources: There are many online resources available that provide tutorials and examples of how to use the tangent function.
Practice problems: Practice problems are a great way to reinforce your understanding of the tangent function.

Conclusion

In conclusion, the tangent function is a fundamental trigonometric function that is used to describe the ratio of the opposite side to the adjacent side in a right-angled triangle. The expression $\tan \left(-\frac{7 \pi}{3}\right)$ is equal to $-\sqrt{3}$ , which is consistent with the properties of the tangent function. We hope that this Q&A article has provided you with a better understanding of the tangent function and its applications.