Evaluate The Expression Below, Expressing Your Answer In Radians.${ 4\pi - 2 \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) }$

Introduction

In this article, we will evaluate the given expression, 4π - 2tan^(-1)(√3/3), and express our answer in radians. The expression involves trigonometric functions, specifically the tangent inverse function, and we will use various mathematical techniques to simplify and evaluate it.

Understanding the Tangent Inverse Function

The tangent inverse function, denoted as tan^(-1)(x), is the inverse of the tangent function. It returns the angle whose tangent is equal to the given value. In other words, if tan(y) = x, then y = tan^(-1)(x). The range of the tangent inverse function is typically restricted to the interval (-π/2, π/2) to ensure a unique output for each input.

Evaluating the Expression

To evaluate the given expression, we can start by simplifying the term inside the tangent inverse function:

${ \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) }$

Using the definition of the tangent function, we can rewrite this as:

${ \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) = \tan^{-1}\left(\tan\left(\frac{\pi}{6}\right)\right) }$

Since the tangent function is periodic with period π, we can rewrite the expression as:

${ \tan^{-1}\left(\tan\left(\frac{\pi}{6}\right)\right) = \frac{\pi}{6} }$

Now, we can substitute this value back into the original expression:

${ 4\pi - 2\left(\frac{\pi}{6}\right) }$

Simplifying the Expression

To simplify the expression, we can start by evaluating the term inside the parentheses:

${ 2\left(\frac{\pi}{6}\right) = \frac{\pi}{3} }$

Now, we can substitute this value back into the original expression:

${ 4\pi - \frac{\pi}{3} }$

Combining Like Terms

To combine like terms, we can start by finding a common denominator for the two terms:

${ 4\pi - \frac{\pi}{3} = \frac{12\pi}{3} - \frac{\pi}{3} }$

Now, we can subtract the two terms:

${ \frac{12\pi}{3} - \frac{\pi}{3} = \frac{11\pi}{3} }$

Conclusion

In this article, we evaluated the given expression, 4π - 2tan^(-1)(√3/3), and expressed our answer in radians. We used various mathematical techniques, including simplifying the term inside the tangent inverse function and combining like terms, to arrive at the final answer. The final answer is:

${ \frac{11\pi}{3} }$

This result can be verified using a calculator or by graphing the expression.

Final Answer

The final answer is $\boxed{\frac{11\pi}{3}}$.

Related Topics

• Trigonometric functions
• Inverse trigonometric functions
• Mathematical techniques for simplifying expressions
• Evaluating expressions in radians

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information and are not specific to the topic of evaluating the given expression.

Introduction

In our previous article, we evaluated the expression 4π - 2tan^(-1)(√3/3) and expressed our answer in radians. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the tangent inverse function?

A: The tangent inverse function, denoted as tan^(-1)(x), is the inverse of the tangent function. It returns the angle whose tangent is equal to the given value. In other words, if tan(y) = x, then y = tan^(-1)(x). The range of the tangent inverse function is typically restricted to the interval (-π/2, π/2) to ensure a unique output for each input.

Q: How do I simplify the term inside the tangent inverse function?

A: To simplify the term inside the tangent inverse function, you can use the definition of the tangent function. In this case, we can rewrite the expression as:

${ \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) = \tan^{-1}\left(\tan\left(\frac{\pi}{6}\right)\right) }$

Since the tangent function is periodic with period π, we can rewrite the expression as:

${ \tan^{-1}\left(\tan\left(\frac{\pi}{6}\right)\right) = \frac{\pi}{6} }$

Q: How do I combine like terms in the expression?

A: To combine like terms, you can start by finding a common denominator for the two terms. In this case, we can rewrite the expression as:

${ 4\pi - \frac{\pi}{3} = \frac{12\pi}{3} - \frac{\pi}{3} }$

Now, we can subtract the two terms:

${ \frac{12\pi}{3} - \frac{\pi}{3} = \frac{11\pi}{3} }$

Q: What is the final answer to the expression?

A: The final answer to the expression is:

${ \frac{11\pi}{3} }$

This result can be verified using a calculator or by graphing the expression.

Q: What are some related topics to this expression?

A: Some related topics to this expression include:

• Trigonometric functions
• Inverse trigonometric functions
• Mathematical techniques for simplifying expressions
• Evaluating expressions in radians

Q: What are some references for further reading?

A: Some references for further reading include:

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Conclusion

In this article, we answered some frequently asked questions related to evaluating the expression 4π - 2tan^(-1)(√3/3). We provided explanations and examples to help clarify the concepts and techniques involved.

Final Answer

The final answer is $\boxed{\frac{11\pi}{3}}$.

Related Topics

• Trigonometric functions
• Inverse trigonometric functions
• Mathematical techniques for simplifying expressions
• Evaluating expressions in radians

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton