Evaluate The Expression Below, Expressing Your Answer In Degrees.$\[ -100^{\circ} + 2 \arctan (1) \\]

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Introduction

In this article, we will delve into the evaluation of a given mathematical expression, focusing on the calculation of the arctan function and its application in trigonometry. The expression in question is ${-100^{\circ} + 2 \arctan (1)}$. Our objective is to simplify this expression and express the result in degrees.

Understanding the Components

Before we proceed with the evaluation, let's break down the components of the expression:

  • Arctan Function: The arctan function, also known as the inverse tangent function, returns the angle whose tangent is a given number. In this case, we have arctan⁡(1)\arctan (1).
  • Angle in Degrees: The expression also involves an angle in degrees, which is −100∘-100^{\circ}.

Evaluating the Arctan Function

The arctan function is a periodic function with a period of π\pi. This means that the arctan function repeats its values every π\pi radians (or 180 degrees). To evaluate arctan⁡(1)\arctan (1), we can use the fact that the tangent function is positive in the first and third quadrants.

Using a calculator or a trigonometric table, we find that arctan⁡(1)=45∘\arctan (1) = 45^{\circ}.

Simplifying the Expression

Now that we have evaluated the arctan function, we can simplify the expression:

−100∘+2arctan⁡(1){-100^{\circ} + 2 \arctan (1)}

Substituting the value of arctan⁡(1)\arctan (1), we get:

−100∘+2⋅45∘{-100^{\circ} + 2 \cdot 45^{\circ}}

Applying the Order of Operations

To simplify the expression further, we need to apply the order of operations (PEMDAS):

  1. Parentheses: None
  2. Exponents: None
  3. Multiplication: 2⋅45∘=90∘2 \cdot 45^{\circ} = 90^{\circ}
  4. Addition: −100∘+90∘-100^{\circ} + 90^{\circ}

Final Evaluation

Now that we have applied the order of operations, we can evaluate the expression:

−100∘+90∘{-100^{\circ} + 90^{\circ}}

Using a calculator or a trigonometric table, we find that the result is −10∘-10^{\circ}.

Conclusion

In this article, we evaluated the given mathematical expression, focusing on the calculation of the arctan function and its application in trigonometry. We simplified the expression by applying the order of operations and evaluated the result in degrees. The final answer is −10∘\boxed{-10^{\circ}}.

Additional Insights

  • Trigonometric Identities: The arctan function is related to the tangent function through the trigonometric identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}. This identity can be used to derive the arctan function from the tangent function.
  • Periodicity: The arctan function is periodic with a period of π\pi. This means that the arctan function repeats its values every π\pi radians (or 180 degrees).
  • Range: The range of the arctan function is (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}). This means that the arctan function returns angles between −π2-\frac{\pi}{2} radians (or −90∘-90^{\circ}) and π2\frac{\pi}{2} radians (or 90∘90^{\circ}).

Final Thoughts

In conclusion, the evaluation of the given mathematical expression involves the calculation of the arctan function and its application in trigonometry. By simplifying the expression and applying the order of operations, we arrived at the final answer of −10∘\boxed{-10^{\circ}}. This article provides a comprehensive analysis of the expression, highlighting the importance of trigonometric identities and the periodicity of the arctan function.

Introduction

In our previous article, we evaluated the expression ${-100^{\circ} + 2 \arctan (1)}$ and arrived at the final answer of −10∘\boxed{-10^{\circ}}. In this article, we will address some common questions and provide additional insights into the evaluation of this expression.

Q&A

Q: What is the arctan function, and how is it related to the tangent function?

A: The arctan function, also known as the inverse tangent function, returns the angle whose tangent is a given number. The arctan function is related to the tangent function through the trigonometric identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}. This identity can be used to derive the arctan function from the tangent function.

Q: Why is the arctan function periodic?

A: The arctan function is periodic with a period of π\pi. This means that the arctan function repeats its values every π\pi radians (or 180 degrees). This periodicity is a result of the trigonometric identity tan⁡(θ+π)=tan⁡(θ)\tan (\theta + \pi) = \tan (\theta).

Q: What is the range of the arctan function?

A: The range of the arctan function is (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}). This means that the arctan function returns angles between −π2-\frac{\pi}{2} radians (or −90∘-90^{\circ}) and π2\frac{\pi}{2} radians (or 90∘90^{\circ}).

Q: How do you evaluate the expression ${-100^{\circ} + 2 \arctan (1)}$?

A: To evaluate the expression, we first need to calculate the value of arctan⁡(1)\arctan (1). Using a calculator or a trigonometric table, we find that arctan⁡(1)=45∘\arctan (1) = 45^{\circ}. Then, we can simplify the expression by applying the order of operations (PEMDAS):

  1. Parentheses: None
  2. Exponents: None
  3. Multiplication: 2⋅45∘=90∘2 \cdot 45^{\circ} = 90^{\circ}
  4. Addition: −100∘+90∘-100^{\circ} + 90^{\circ}

Q: What is the final answer to the expression ${-100^{\circ} + 2 \arctan (1)}$?

A: The final answer to the expression is −10∘\boxed{-10^{\circ}}.

Additional Insights

  • Trigonometric Identities: The arctan function is related to the tangent function through the trigonometric identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}. This identity can be used to derive the arctan function from the tangent function.
  • Periodicity: The arctan function is periodic with a period of π\pi. This means that the arctan function repeats its values every π\pi radians (or 180 degrees).
  • Range: The range of the arctan function is (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}). This means that the arctan function returns angles between −π2-\frac{\pi}{2} radians (or −90∘-90^{\circ}) and π2\frac{\pi}{2} radians (or 90∘90^{\circ}).

Final Thoughts

In conclusion, the evaluation of the expression ${-100^{\circ} + 2 \arctan (1)}$ involves the calculation of the arctan function and its application in trigonometry. By simplifying the expression and applying the order of operations, we arrived at the final answer of −10∘\boxed{-10^{\circ}}. This article provides a comprehensive analysis of the expression, highlighting the importance of trigonometric identities and the periodicity of the arctan function.

Related Questions

  • What is the difference between the arctan function and the tangent function?
  • How do you evaluate the expression ${-100^{\circ} + 2 \arctan (1)}$ using a calculator?
  • What is the range of the arctan function in degrees?
  • How do you apply the order of operations to simplify the expression ${-100^{\circ} + 2 \arctan (1)}$?

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