What Is The Approximate Value Of $\tan \left(\frac{3 \pi}{5}\right)$?A. -3.078 B. 0.033 C. 1.083 D. 2.468

Introduction

In this article, we will explore the concept of trigonometric functions, specifically the tangent function, and how to find the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$. The tangent function is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science.

Understanding the Tangent Function

The tangent function is a trigonometric function that is defined as the ratio of the sine and cosine functions. It is denoted by the symbol $\tan(x)$ and is defined as:

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

The tangent function has a periodic nature, with a period of $\pi$. This means that the value of the tangent function repeats every $\pi$ radians.

Finding the Approximate Value of $\tan \left(\frac{3 \pi}{5}\right)$

To find the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$, we can use a calculator or a computer program to evaluate the expression. However, in this article, we will use a mathematical approach to find the approximate value.

First, we need to find the values of $\sin \left(\frac{3 \pi}{5}\right)$ and $\cos \left(\frac{3 \pi}{5}\right)$. We can use the unit circle or a trigonometric identity to find these values.

Using the unit circle, we can find that:

$$\sin \left(\frac{3 \pi}{5}\right) = \frac{\sqrt{5} - 1}{4}$$

$$\cos \left(\frac{3 \pi}{5}\right) = \frac{\sqrt{5} + 1}{4}$$

Now, we can use these values to find the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$:

$$\tan \left(\frac{3 \pi}{5}\right) = \frac{\sin \left(\frac{3 \pi}{5}\right)}{\cos \left(\frac{3 \pi}{5}\right)} = \frac{\frac{\sqrt{5} - 1}{4}}{\frac{\sqrt{5} + 1}{4}}$$

Simplifying this expression, we get:

$$\tan \left(\frac{3 \pi}{5}\right) = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$$

Simplifying the Expression

To simplify the expression, we can use the conjugate of the denominator to rationalize the expression. The conjugate of $\sqrt{5} + 1$ is $\sqrt{5} - 1$.

Multiplying the numerator and denominator by the conjugate, we get:

$$\tan \left(\frac{3 \pi}{5}\right) = \frac{(\sqrt{5} - 1)(\sqrt{5} - 1)}{(\sqrt{5} + 1)(\sqrt{5} - 1)}$$

Simplifying this expression, we get:

$$\tan \left(\frac{3 \pi}{5}\right) = \frac{5 - 2\sqrt{5} + 1}{5 - 1}$$

$$\tan \left(\frac{3 \pi}{5}\right) = \frac{6 - 2\sqrt{5}}{4}$$

Approximating the Value

To approximate the value of $\tan \left(\frac{3 \pi}{5}\right)$, we can use a calculator or a computer program to evaluate the expression. However, in this article, we will use a mathematical approach to approximate the value.

Using the approximate value of $\sqrt{5} \approx 2.236$, we can substitute this value into the expression:

$$\tan \left(\frac{3 \pi}{5}\right) \approx \frac{6 - 2(2.236)}{4}$$

Simplifying this expression, we get:

$$\tan \left(\frac{3 \pi}{5}\right) \approx \frac{6 - 4.472}{4}$$

$$\tan \left(\frac{3 \pi}{5}\right) \approx \frac{1.528}{4}$$

$$\tan \left(\frac{3 \pi}{5}\right) \approx 0.382$$

Conclusion

In this article, we explored the concept of trigonometric functions, specifically the tangent function, and how to find the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$. We used a mathematical approach to find the approximate value, and we approximated the value using a calculator or a computer program.

The approximate value of $\tan \left(\frac{3 \pi}{5}\right)$ is:

$$\tan \left(\frac{3 \pi}{5}\right) \approx 0.382$$

This value is closest to option B. 0.033.

Final Answer

The final answer is B. 0.033.

Introduction

In our previous article, we explored the concept of trigonometric functions, specifically the tangent function, and how to find the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$. We used a mathematical approach to find the approximate value, and we approximated the value using a calculator or a computer program.

In this article, we will answer some frequently asked questions about the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$.

Q1: What is the tangent function?

A1: The tangent function is a trigonometric function that is defined as the ratio of the sine and cosine functions. It is denoted by the symbol $\tan(x)$ and is defined as:

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

Q2: How do I find the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$?

A2: To find the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$, you can use a calculator or a computer program to evaluate the expression. Alternatively, you can use a mathematical approach to find the approximate value, as we did in our previous article.

Q3: What is the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$?

A3: The approximate value of $\tan \left(\frac{3 \pi}{5}\right)$ is:

$$\tan \left(\frac{3 \pi}{5}\right) \approx 0.382$$

Q4: How do I simplify the expression $\tan \left(\frac{3 \pi}{5}\right) = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$?

A4: To simplify the expression, you can use the conjugate of the denominator to rationalize the expression. The conjugate of $\sqrt{5} + 1$ is $\sqrt{5} - 1$.

Multiplying the numerator and denominator by the conjugate, you get:

$$\tan \left(\frac{3 \pi}{5}\right) = \frac{(\sqrt{5} - 1)(\sqrt{5} - 1)}{(\sqrt{5} + 1)(\sqrt{5} - 1)}$$

Simplifying this expression, you get:

$$\tan \left(\frac{3 \pi}{5}\right) = \frac{5 - 2\sqrt{5} + 1}{5 - 1}$$

$$\tan \left(\frac{3 \pi}{5}\right) = \frac{6 - 2\sqrt{5}}{4}$$

Q5: What is the final answer?

A5: The final answer is B. 0.033.

Conclusion

In this article, we answered some frequently asked questions about the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$. We provided the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$ and explained how to simplify the expression.

We hope that this article has been helpful in answering your questions about the approximate value of $\tan \left(\frac{3 \pi}{5}\right)$. If you have any further questions, please don't hesitate to ask.

Final Answer

The final answer is B. 0.033.