Evaluate The Expression Below, Expressing Your Answer In Degrees.$200^{\circ} - 2 \arctan \left(-\frac{\sqrt{3}}{3}\right$\]

Introduction

In this article, we will evaluate the given expression, $200^{\circ} - 2 \arctan \left(-\frac{\sqrt{3}}{3}\right)$, and express our answer in degrees. This involves understanding the properties of trigonometric functions, specifically the arctan function, and how to manipulate them to simplify the expression.

Understanding the Arctan Function

The arctan function, denoted as $\arctan x$, is the inverse of the tangent function. It returns the angle whose tangent is a given number. In other words, if $\tan \theta = x$, then $\arctan x = \theta$. The range of the arctan function is $(-\frac{\pi}{2}, \frac{\pi}{2})$, which is equivalent to $(-90^{\circ}, 90^{\circ})$.

Evaluating the Expression

To evaluate the expression $200^{\circ} - 2 \arctan \left(-\frac{\sqrt{3}}{3}\right)$, we need to first simplify the arctan term. We can do this by using the properties of the arctan function.

Simplifying the Arctan Term

We can rewrite the arctan term as follows:

$$\arctan \left(-\frac{\sqrt{3}}{3}\right) = \arctan \left(\frac{\sqrt{3}}{3}\right)$$

This is because the arctan function is an odd function, meaning that $\arctan (-x) = -\arctan x$.

Using the Arctan Identity

We can use the arctan identity $\arctan x = \arctan \left(\frac{x}{1}\right)$ to rewrite the arctan term as follows:

$$\arctan \left(\frac{\sqrt{3}}{3}\right) = \arctan \left(\frac{\sqrt{3}}{3+0}\right)$$

This simplifies to:

$$\arctan \left(\frac{\sqrt{3}}{3}\right) = \arctan \left(\frac{\sqrt{3}}{3}\right)$$

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Introduction

In our previous article, we evaluated the expression $200^{\circ} - 2 \arctan \left(-\frac{\sqrt{3}}{3}\right)$ and simplified the arctan term. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the arctan function and how does it relate to the tangent function?

A: The arctan function, denoted as $\arctan x$, is the inverse of the tangent function. It returns the angle whose tangent is a given number. In other words, if $\tan \theta = x$, then $\arctan x = \theta$. The range of the arctan function is $(-\frac{\pi}{2}, \frac{\pi}{2})$, which is equivalent to $(-90^{\circ}, 90^{\circ})$.

Q: How do you simplify the arctan term in the expression?

A: To simplify the arctan term, we can use the properties of the arctan function. We can rewrite the arctan term as follows:

$$\arctan \left(-\frac{\sqrt{3}}{3}\right) = \arctan \left(\frac{\sqrt{3}}{3}\right)$$

This is because the arctan function is an odd function, meaning that $\arctan (-x) = -\arctan x$.

Q: What is the arctan identity and how is it used to simplify the arctan term?

A: The arctan identity is $\arctan x = \arctan \left(\frac{x}{1}\right)$. We can use this identity to rewrite the arctan term as follows:

$$\arctan \left(\frac{\sqrt{3}}{3}\right) = \arctan \left(\frac{\sqrt{3}}{3+0}\right)$$

This simplifies to:

$$\arctan \left(\frac{\sqrt{3}}{3}\right) = \arctan \left(\frac{\sqrt{3}}{3}\right)$$

Q: How do you evaluate the expression $200^{\circ} - 2 \arctan \left(-\frac{\sqrt{3}}{3}\right)$?

A: To evaluate the expression, we need to first simplify the arctan term. We can do this by using the properties of the arctan function, as described above. Once we have simplified the arctan term, we can subtract it from $200^{\circ}$ to get the final answer.

Q: What is the final answer to the expression $200^{\circ} - 2 \arctan \left(-\frac{\sqrt{3}}{3}\right)$?

A: The final answer to the expression is $120^{\circ}$. This is because the arctan term simplifies to $60^{\circ}$, and subtracting this from $200^{\circ}$ gives us $120^{\circ}$.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional information on the topic of evaluating the expression $200^{\circ} - 2 \arctan \left(-\frac{\sqrt{3}}{3}\right)$. We hope that this article has been helpful in understanding the properties of the arctan function and how to simplify the expression.

Additional Resources

• Arctan Function
• Trigonometric Functions
• Mathematical Properties of the Arctan Function

Final Answer

The final answer to the expression $200^{\circ} - 2 \arctan \left(-\frac{\sqrt{3}}{3}\right)$ is $120^{\circ}$.