What Is The Exact Value Of \csc \left(\frac{16 \pi}{3}\right ]?A. − 2 3 3 -\frac{2 \sqrt{3}}{3} − 3 2 3 ​ ​ B. 2 3 3 \frac{2 \sqrt{3}}{3} 3 2 3 ​ ​ C. − 2 -\sqrt{2} − 2 ​ D. 2 \sqrt{2} 2 ​

Introduction

In trigonometry, the cosecant function is the reciprocal of the sine function. It is denoted by $\csc$ and is defined as $\csc(\theta) = \frac{1}{\sin(\theta)}$. The cosecant function is periodic with a period of $2\pi$, which means that the value of $\csc(\theta)$ repeats every $2\pi$ radians. In this article, we will explore the exact value of $\csc \left(\frac{16 \pi}{3}\right)$.

Understanding the Problem

To find the exact value of $\csc \left(\frac{16 \pi}{3}\right)$, we need to first understand the concept of trigonometric functions and their periodicity. The sine function is periodic with a period of $2\pi$, which means that $\sin(\theta) = \sin(\theta + 2\pi)$ for any angle $\theta$. Similarly, the cosecant function is also periodic with a period of $2\pi$.

Simplifying the Angle

The given angle is $\frac{16 \pi}{3}$. To simplify this angle, we can use the fact that $\frac{16 \pi}{3} = \frac{16 \pi}{3} - 4 \pi + 4 \pi = \frac{4 \pi}{3} + 4 \pi$. This simplification allows us to rewrite the angle in terms of a more familiar angle.

Using Trigonometric Identities

To find the exact value of $\csc \left(\frac{16 \pi}{3}\right)$, we can use the trigonometric identity $\csc(\theta) = \frac{1}{\sin(\theta)}$. We can also use the fact that $\sin(\theta) = \sin(\theta + 2\pi)$ for any angle $\theta$. By using these identities, we can simplify the expression and find the exact value of $\csc \left(\frac{16 \pi}{3}\right)$.

Finding the Exact Value

Using the trigonometric identity $\csc(\theta) = \frac{1}{\sin(\theta)}$, we can rewrite the expression as $\csc \left(\frac{16 \pi}{3}\right) = \frac{1}{\sin \left(\frac{16 \pi}{3}\right)}$. We can simplify the angle $\frac{16 \pi}{3}$ by using the fact that $\frac{16 \pi}{3} = \frac{4 \pi}{3} + 4 \pi$. This allows us to rewrite the expression as $\csc \left(\frac{16 \pi}{3}\right) = \frac{1}{\sin \left(\frac{4 \pi}{3}\right)}$.

Using the Unit Circle

To find the exact value of $\sin \left(\frac{4 \pi}{3}\right)$, we can use the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine function is defined as the y-coordinate of a point on the unit circle. By using the unit circle, we can find the exact value of $\sin \left(\frac{4 \pi}{3}\right)$.

Finding the Exact Value of $\sin \left(\frac{4 \pi}{3}\right)$

Using the unit circle, we can find the exact value of $\sin \left(\frac{4 \pi}{3}\right)$. The point on the unit circle corresponding to the angle $\frac{4 \pi}{3}$ has coordinates $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$. Therefore, the exact value of $\sin \left(\frac{4 \pi}{3}\right)$ is $-\frac{\sqrt{3}}{2}$.

Finding the Exact Value of $\csc \left(\frac{16 \pi}{3}\right)$

Using the trigonometric identity $\csc(\theta) = \frac{1}{\sin(\theta)}$, we can rewrite the expression as $\csc \left(\frac{16 \pi}{3}\right) = \frac{1}{\sin \left(\frac{4 \pi}{3}\right)}$. We have found that the exact value of $\sin \left(\frac{4 \pi}{3}\right)$ is $-\frac{\sqrt{3}}{2}$. Therefore, the exact value of $\csc \left(\frac{16 \pi}{3}\right)$ is $\frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2 \sqrt{3}}{3}$.

Conclusion

In this article, we have explored the exact value of $\csc \left(\frac{16 \pi}{3}\right)$. We have used trigonometric identities and the unit circle to simplify the expression and find the exact value. The exact value of $\csc \left(\frac{16 \pi}{3}\right)$ is $-\frac{2 \sqrt{3}}{3}$.

Final Answer

The final answer is $\boxed{-\frac{2 \sqrt{3}}{3}}$.

Introduction

In our previous article, we explored the exact value of $\csc \left(\frac{16 \pi}{3}\right)$. We used trigonometric identities and the unit circle to simplify the expression and find the exact value. In this article, we will answer some frequently asked questions related to the exact value of $\csc \left(\frac{16 \pi}{3}\right)$.

Q1: What is the period of the cosecant function?

A1: The period of the cosecant function is $2\pi$, which means that the value of $\csc(\theta)$ repeats every $2\pi$ radians.

Q2: How do you simplify the angle $\frac{16 \pi}{3}$?

A2: To simplify the angle $\frac{16 \pi}{3}$, we can use the fact that $\frac{16 \pi}{3} = \frac{4 \pi}{3} + 4 \pi$. This allows us to rewrite the angle in terms of a more familiar angle.

Q3: What is the exact value of $\sin \left(\frac{4 \pi}{3}\right)$?

A3: The exact value of $\sin \left(\frac{4 \pi}{3}\right)$ is $-\frac{\sqrt{3}}{2}$. This can be found using the unit circle.

Q4: How do you find the exact value of $\csc \left(\frac{16 \pi}{3}\right)$?

A4: To find the exact value of $\csc \left(\frac{16 \pi}{3}\right)$, we can use the trigonometric identity $\csc(\theta) = \frac{1}{\sin(\theta)}$. We can also use the fact that $\sin(\theta) = \sin(\theta + 2\pi)$ for any angle $\theta$.

Q5: What is the final answer to the problem?

A5: The final answer to the problem is $\boxed{-\frac{2 \sqrt{3}}{3}}$.

Q6: Why is the cosecant function important in trigonometry?

A6: The cosecant function is important in trigonometry because it is the reciprocal of the sine function. It is used to solve problems involving right triangles and is an essential concept in trigonometry.

Q7: Can you provide more examples of how to use the cosecant function?

A7: Yes, here are a few more examples of how to use the cosecant function:

• $\csc(0) = 1$
• $\csc(\frac{\pi}{2}) = 1$
• $\csc(\pi) = -1$
• $\csc(\frac{3\pi}{2}) = -1$

Q8: How do you graph the cosecant function?

A8: To graph the cosecant function, we can use the unit circle and the fact that the cosecant function is the reciprocal of the sine function. We can also use a graphing calculator to visualize the graph of the cosecant function.

Q9: What are some common applications of the cosecant function?

A9: The cosecant function has many common applications in trigonometry, including:

• Solving problems involving right triangles
• Finding the length of sides of triangles
• Determining the measure of angles in triangles
• Solving problems involving circular functions

Q10: Can you provide more resources for learning about the cosecant function?

A10: Yes, here are a few more resources for learning about the cosecant function:

• Khan Academy: Trigonometry
• Mathway: Cosecant Function
• Wolfram Alpha: Cosecant Function

Conclusion

In this article, we have answered some frequently asked questions related to the exact value of $\csc \left(\frac{16 \pi}{3}\right)$. We have used trigonometric identities and the unit circle to simplify the expression and find the exact value. We have also provided more examples of how to use the cosecant function and discussed some common applications of the cosecant function.