Evaluate The Expression: $\tan^{-1}(-\sqrt{3}$\]

Introduction

In mathematics, the inverse tangent function, denoted as $\tan^{-1}$, is a fundamental concept in trigonometry. It is used to find the angle whose tangent is a given value. In this article, we will evaluate the expression $\tan^{-1}(-\sqrt{3})$ and explore its properties.

Understanding the Inverse Tangent Function

The inverse tangent function, $\tan^{-1}$, is defined as the angle whose tangent is a given value. In other words, if $\tan(\theta) = x$, then $\tan^{-1}(x) = \theta$. The range of the inverse tangent function is typically restricted to the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$ to ensure a unique solution.

Evaluating the Expression

To evaluate the expression $\tan^{-1}(-\sqrt{3})$, we need to find the angle whose tangent is $-\sqrt{3}$. We know that $\tan(\frac{\pi}{3}) = \sqrt{3}$, so we can use this information to find the angle whose tangent is $-\sqrt{3}$.

Using the Properties of the Tangent Function

The tangent function has a property that $\tan(-\theta) = -\tan(\theta)$. Using this property, we can rewrite the expression $\tan^{-1}(-\sqrt{3})$ as $\tan^{-1}(\sqrt{3})$.

Finding the Angle

We know that $\tan(\frac{\pi}{3}) = \sqrt{3}$, so we can conclude that $\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$. However, we need to consider the range of the inverse tangent function, which is restricted to the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$. Since $\frac{\pi}{3}$ is greater than $\frac{\pi}{2}$, we need to find an equivalent angle within the range.

Using the Periodicity of the Tangent Function

The tangent function has a periodicity of $\pi$, meaning that $\tan(\theta + \pi) = \tan(\theta)$. Using this property, we can find an equivalent angle within the range by subtracting $\pi$ from $\frac{\pi}{3}$.

Finding the Equivalent Angle

$\tan^{-1}(\sqrt{3}) = \frac{\pi}{3} - \pi = -\frac{2\pi}{3}$

Conclusion

In conclusion, the expression $\tan^{-1}(-\sqrt{3})$ evaluates to $-\frac{2\pi}{3}$. This result is obtained by using the properties of the tangent function, including its periodicity and the range of the inverse tangent function.

Properties of the Inverse Tangent Function

The inverse tangent function has several properties that are useful in evaluating expressions. Some of these properties include:

• Range: The range of the inverse tangent function is typically restricted to the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$.
• Periodicity: The inverse tangent function has a periodicity of $\pi$, meaning that $\tan^{-1}(\tan(\theta + \pi)) = \tan^{-1}(\tan(\theta)) + \pi$.
• Symmetry: The inverse tangent function is symmetric about the origin, meaning that $\tan^{-1}(-x) = -\tan^{-1}(x)$.

Applications of the Inverse Tangent Function

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

• Trigonometry: The inverse tangent function is used to find the angle whose tangent is a given value.
• Calculus: The inverse tangent function is used to find the derivative of trigonometric functions.
• Engineering: The inverse tangent function is used to find the angle of elevation or depression in problems involving right triangles.

Final Thoughts

In conclusion, the expression $\tan^{-1}(-\sqrt{3})$ evaluates to $-\frac{2\pi}{3}$. This result is obtained by using the properties of the tangent function, including its periodicity and the range of the inverse tangent function. The inverse tangent function has several properties and applications that make it a fundamental concept in mathematics and science.

Introduction

In our previous article, we evaluated the expression $\tan^{-1}(-\sqrt{3})$ and found that it equals $-\frac{2\pi}{3}$. In this article, we will answer some frequently asked questions related to this expression.

Q: What is the inverse tangent function?

A: The inverse tangent function, denoted as $\tan^{-1}$, is a mathematical function that returns the angle whose tangent is a given value. In other words, if $\tan(\theta) = x$, then $\tan^{-1}(x) = \theta$.

Q: What is the range of the inverse tangent function?

A: The range of the inverse tangent function is typically restricted to the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$ to ensure a unique solution.

Q: How do I evaluate the expression $\tan^{-1}(-\sqrt{3})$?

A: To evaluate the expression $\tan^{-1}(-\sqrt{3})$, you need to find the angle whose tangent is $-\sqrt{3}$. You can use the properties of the tangent function, including its periodicity and the range of the inverse tangent function, to find the solution.

Q: What is the periodicity of the inverse tangent function?

A: The inverse tangent function has a periodicity of $\pi$, meaning that $\tan^{-1}(\tan(\theta + \pi)) = \tan^{-1}(\tan(\theta)) + \pi$.

Q: How do I use the periodicity of the inverse tangent function to evaluate the expression $\tan^{-1}(-\sqrt{3})$?

A: To use the periodicity of the inverse tangent function, you can rewrite the expression $\tan^{-1}(-\sqrt{3})$ as $\tan^{-1}(\sqrt{3})$ and then find an equivalent angle within the range by subtracting $\pi$ from $\frac{\pi}{3}$.

Q: What is the equivalent angle of $\frac{\pi}{3}$ within the range of the inverse tangent function?

A: The equivalent angle of $\frac{\pi}{3}$ within the range of the inverse tangent function is $-\frac{2\pi}{3}$.

Q: What are some applications of the inverse tangent function?

A: The inverse tangent function has several applications in mathematics and science, including:

• Trigonometry: The inverse tangent function is used to find the angle whose tangent is a given value.
• Calculus: The inverse tangent function is used to find the derivative of trigonometric functions.
• Engineering: The inverse tangent function is used to find the angle of elevation or depression in problems involving right triangles.

Q: What are some common mistakes to avoid when evaluating the expression $\tan^{-1}(-\sqrt{3})$?

A: Some common mistakes to avoid when evaluating the expression $\tan^{-1}(-\sqrt{3})$ include:

• Not considering the range of the inverse tangent function: Make sure to consider the range of the inverse tangent function when evaluating the expression.
• Not using the periodicity of the inverse tangent function: Use the periodicity of the inverse tangent function to find an equivalent angle within the range.
• Not simplifying the expression: Simplify the expression by using the properties of the tangent function.

Conclusion

In conclusion, the expression $\tan^{-1}(-\sqrt{3})$ evaluates to $-\frac{2\pi}{3}$. This result is obtained by using the properties of the tangent function, including its periodicity and the range of the inverse tangent function. We hope that this Q&A article has helped to clarify any questions you may have had about this expression.