What Is The Exact Value Of $\sec \frac{\pi}{6}$?A. $\sqrt{2}$B. $\frac{2 \sqrt{3}}{3}$C. 2D. 0

Introduction

In trigonometry, the secant function is the reciprocal of the cosine function. It is defined as the ratio of the length of the hypotenuse to the length of the adjacent side in a right-angled triangle. The secant function is denoted by the symbol sec and is used to calculate the length of the hypotenuse in a right-angled triangle. In this article, we will explore the exact value of $\sec \frac{\pi}{6}$.

Understanding the Secant Function

The secant function is defined as the reciprocal of the cosine function. It is denoted by the symbol sec and is used to calculate the length of the hypotenuse in a right-angled triangle. The secant function can be expressed as:

$$\sec \theta = \frac{1}{\cos \theta}$$

where $\theta$ is the angle in a right-angled triangle.

Calculating the Exact Value of $\sec \frac{\pi}{6}$

To calculate the exact value of $\sec \frac{\pi}{6}$, we need to use the definition of the secant function and the value of the cosine function for the angle $\frac{\pi}{6}$.

The cosine function for the angle $\frac{\pi}{6}$ is given by:

$$\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$$

Using the definition of the secant function, we can calculate the exact value of $\sec \frac{\pi}{6}$ as follows:

$$\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$$

Simplifying the Exact Value

To simplify the exact value of $\sec \frac{\pi}{6}$, we can rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{3}$.

$$\sec \frac{\pi}{6} = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$$

Conclusion

In conclusion, the exact value of $\sec \frac{\pi}{6}$ is $\frac{2\sqrt{3}}{3}$. This value can be obtained by using the definition of the secant function and the value of the cosine function for the angle $\frac{\pi}{6}$.

Final Answer

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

Discussion

The discussion category for this article is mathematics. The article explores the exact value of $\sec \frac{\pi}{6}$ and provides a step-by-step solution to calculate the value.

Related Articles

• What is the exact value of $\tan \frac{\pi}{6}$?
• What is the exact value of $\sin \frac{\pi}{6}$?
• What is the exact value of $\cos \frac{\pi}{6}$?

References

• [1] Trigonometry, 10th edition, by Charles P. McKeague and Mark D. Turner
• [2] Calculus, 8th edition, by Michael Spivak
• [3] Mathematics, 12th edition, by Michael Artin
  

Introduction

In our previous article, we explored the exact value of $\sec \frac{\pi}{6}$. In this article, we will answer some frequently asked questions related to the exact value of $\sec \frac{\pi}{6}$.

Q&A

Q1: What is the definition of the secant function?

A1: The secant function is the reciprocal of the cosine function. It is defined as the ratio of the length of the hypotenuse to the length of the adjacent side in a right-angled triangle.

Q2: How do you calculate the exact value of $\sec \frac{\pi}{6}$?

A2: To calculate the exact value of $\sec \frac\pi}{6}$, you need to use the definition of the secant function and the value of the cosine function for the angle $\frac{\pi}{6}$. The cosine function for the angle $\frac{\pi}{6}$ is given by $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$. Using the definition of the secant function, we can calculate the exact value of $\sec \frac{\pi}{6}$ as follows $\sec \frac{\pi{6} = \frac{1}{\cos \frac{\pi}{6}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$

Q3: How do you simplify the exact value of $\sec \frac{\pi}{6}$?

A3: To simplify the exact value of $\sec \frac\pi}{6}$, you can rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{3}$. This gives us $\sec \frac{\pi{6} = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

Q4: What is the final answer for the exact value of $\sec \frac{\pi}{6}$?

A4: The final answer for the exact value of $\sec \frac{\pi}{6}$ is $\frac{2\sqrt{3}}{3}$.

Q5: What is the discussion category for this article?

A5: The discussion category for this article is mathematics.

Q6: What are some related articles to this topic?

A6: Some related articles to this topic include:

• What is the exact value of $\tan \frac{\pi}{6}$?
• What is the exact value of $\sin \frac{\pi}{6}$?
• What is the exact value of $\cos \frac{\pi}{6}$?

Conclusion

In conclusion, we have answered some frequently asked questions related to the exact value of $\sec \frac{\pi}{6}$. We hope that this article has provided you with a better understanding of the exact value of $\sec \frac{\pi}{6}$ and its applications in mathematics.

Final Answer

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

Discussion

The discussion category for this article is mathematics. The article explores the exact value of $\sec \frac{\pi}{6}$ and provides a step-by-step solution to calculate the value.

Related Articles

• What is the exact value of $\tan \frac{\pi}{6}$?
• What is the exact value of $\sin \frac{\pi}{6}$?
• What is the exact value of $\cos \frac{\pi}{6}$?

References

• [1] Trigonometry, 10th edition, by Charles P. McKeague and Mark D. Turner
• [2] Calculus, 8th edition, by Michael Spivak
• [3] Mathematics, 12th edition, by Michael Artin