Evaluate The Following Expression:$\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) =$

Introduction

In this article, we will evaluate the given expression $\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) =$. This expression involves trigonometric functions, specifically the secant and inverse secant functions. We will break down the expression step by step and use trigonometric identities to simplify it.

Understanding the Expression

The expression $\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right)$ involves two main functions: the secant function and the inverse secant function. The secant function is defined as the reciprocal of the cosine function, i.e., $\sec(x) = \frac{1}{\cos(x)}$. The inverse secant function, also known as the arcsecant function, is the inverse of the secant function.

Evaluating the Inverse Secant Function

To evaluate the expression, we first need to find the value of $\operatorname{arcsec}\left(\frac{9}{5}\right)$. The inverse secant function returns an angle whose secant is equal to the given value. In this case, we need to find the angle whose secant is equal to $\frac{9}{5}$.

Using Trigonometric Identities

We can use the Pythagorean identity $\sec^2(x) - \tan^2(x) = 1$ to rewrite the expression. Since $\sec(x) = \frac{1}{\cos(x)}$, we can rewrite the expression as $\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) = \frac{1}{\cos \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right)}$.

Simplifying the Expression

Using the definition of the inverse secant function, we know that $\cos \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) = \frac{5}{9}$. Therefore, we can simplify the expression as follows:

$\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) = \frac{1}{\cos \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right)} = \frac{1}{\frac{5}{9}} = \frac{9}{5}$

Conclusion

In conclusion, we have evaluated the given expression $\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) =$. We used trigonometric identities and the definition of the inverse secant function to simplify the expression and find its value.

Frequently Asked Questions

• What is the secant function? The secant function is defined as the reciprocal of the cosine function, i.e., $\sec(x) = \frac{1}{\cos(x)}$.
• What is the inverse secant function? The inverse secant function, also known as the arcsecant function, is the inverse of the secant function.
• How do you evaluate the inverse secant function? To evaluate the inverse secant function, you need to find the angle whose secant is equal to the given value.

Final Answer

The final answer is: $\boxed{\frac{9}{5}}$

Introduction

In our previous article, we evaluated the expression $\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) =$. We used trigonometric identities and the definition of the inverse secant function to simplify the expression and find its value. In this article, we will answer some frequently asked questions related to evaluating trigonometric expressions.

Q&A

Q: What is the secant function?

A: The secant function is defined as the reciprocal of the cosine function, i.e., $\sec(x) = \frac{1}{\cos(x)}$.

Q: What is the inverse secant function?

A: The inverse secant function, also known as the arcsecant function, is the inverse of the secant function.

Q: How do you evaluate the inverse secant function?

A: To evaluate the inverse secant function, you need to find the angle whose secant is equal to the given value.

Q: What is the difference between the secant and cosine functions?

A: The secant function is the reciprocal of the cosine function, i.e., $\sec(x) = \frac{1}{\cos(x)}$. This means that the secant function is always positive, while the cosine function can be positive or negative.

Q: How do you simplify trigonometric expressions involving the secant function?

A: To simplify trigonometric expressions involving the secant function, you can use trigonometric identities, such as the Pythagorean identity $\sec^2(x) - \tan^2(x) = 1$.

Q: What is the relationship between the secant and tangent functions?

A: The secant and tangent functions are related by the Pythagorean identity $\sec^2(x) - \tan^2(x) = 1$. This means that the secant function can be expressed in terms of the tangent function.

Q: How do you evaluate trigonometric expressions involving the inverse secant function?

A: To evaluate trigonometric expressions involving the inverse secant function, you need to find the angle whose secant is equal to the given value. This can be done using trigonometric identities and the definition of the inverse secant function.

Examples

Example 1: Evaluating the Secant Function

Evaluate the expression $\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right)$.

Solution: Using the definition of the inverse secant function, we know that $\cos \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) = \frac{5}{9}$. Therefore, we can simplify the expression as follows:

$\sec \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right) = \frac{1}{\cos \left(\operatorname{arcsec}\left(\frac{9}{5}\right)\right)} = \frac{1}{\frac{5}{9}} = \frac{9}{5}$

Example 2: Evaluating the Inverse Secant Function

Evaluate the expression $\operatorname{arcsec}\left(\frac{9}{5}\right)$.

Solution: To evaluate the inverse secant function, we need to find the angle whose secant is equal to $\frac{9}{5}$. Using the definition of the secant function, we know that $\sec(x) = \frac{1}{\cos(x)}$. Therefore, we can rewrite the expression as follows:

$\operatorname{arcsec}\left(\frac{9}{5}\right) = \cos^{-1}\left(\frac{5}{9}\right)$

Conclusion

In conclusion, we have answered some frequently asked questions related to evaluating trigonometric expressions. We have discussed the secant and inverse secant functions, and provided examples of how to evaluate these functions.

Frequently Asked Questions

• What is the secant function?
• What is the inverse secant function?
• How do you evaluate the inverse secant function?
• What is the difference between the secant and cosine functions?
• How do you simplify trigonometric expressions involving the secant function?
• What is the relationship between the secant and tangent functions?
• How do you evaluate trigonometric expressions involving the inverse secant function?

Final Answer

The final answer is: $\boxed{\frac{9}{5}}$