Evaluate \[$\operatorname{arcsec}(-\sqrt{2})\$\].

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Introduction

In mathematics, the inverse secant function, denoted as $\operatorname{arcsec}$, is the inverse of the secant function. It is a multivalued function, but we usually restrict its range to the interval $[0, \pi]$ to make it a single-valued function. The secant function is defined as the reciprocal of the cosine function, and the inverse secant function is defined as the angle whose secant is a given value. In this article, we will evaluate the expression $\operatorname{arcsec}(-\sqrt{2})$.

Understanding the Secant Function

The secant function is defined as the reciprocal of the cosine function. It is an odd function, meaning that $\sec(-x) = -\sec(x)$ for all $x$ in the domain of the function. The range of the secant function is all real numbers except for $-1$ and $1$. The graph of the secant function is a periodic function with a period of $2\pi$.

Understanding the Inverse Secant Function

The inverse secant function, denoted as $\operatorname{arcsec}$, is the inverse of the secant function. It is a multivalued function, but we usually restrict its range to the interval $[0, \pi]$ to make it a single-valued function. The inverse secant function is defined as the angle whose secant is a given value. The range of the inverse secant function is the interval $[0, \pi]$.

Evaluating $\operatorname{arcsec}(-\sqrt{2})$

To evaluate the expression $\operatorname{arcsec}(-\sqrt{2})$, we need to find the angle whose secant is $-\sqrt{2}$. We can start by finding the angle whose secant is $\sqrt{2}$, which is $\frac{\pi}{4}$.

Using the Identity $\sec^2(x) = 1 + \tan^2(x)$

We can use the identity $\sec^2(x) = 1 + \tan^2(x)$ to find the angle whose secant is $-\sqrt{2}$. We have:

$$\sec^2(x) = 1 + \tan^2(x)$$

$$\sec^2(x) = 1 + \left(\frac{\sqrt{2}}{1}\right)^2$$

$$\sec^2(x) = 1 + 2$$

$$\sec^2(x) = 3$$

$$\sec(x) = \pm \sqrt{3}$$

Since we are looking for the angle whose secant is $-\sqrt{2}$, we take the negative square root:

$$\sec(x) = -\sqrt{3}$$

Using the Inverse Secant Function

We can use the inverse secant function to find the angle whose secant is $-\sqrt{3}$:

$$x = \operatorname{arcsec}(-\sqrt{3})$$

Using the Identity $\sec(x) = \frac{1}{\cos(x)}$

We can use the identity $\sec(x) = \frac{1}{\cos(x)}$ to find the angle whose secant is $-\sqrt{3}$:

$$\sec(x) = \frac{1}{\cos(x)}$$

$$-\sqrt{3} = \frac{1}{\cos(x)}$$

$$\cos(x) = -\frac{1}{\sqrt{3}}$$

Using the Inverse Cosine Function

We can use the inverse cosine function to find the angle whose cosine is $-\frac{1}{\sqrt{3}}$:

$$x = \cos^{-1}\left(-\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x)$

We can use the identity $\cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x)$ to find the angle whose cosine is $-\frac{1}{\sqrt{3}}$:

$$x = \frac{\pi}{2} - \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$

We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$

We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$

We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$

We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$

We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$

We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$

We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$

We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Using the Identity {{content}}lt;br/>

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Q: What is the inverse secant function?

A: The inverse secant function, denoted as $\operatorname{arcsec}$, is the inverse of the secant function. It is a multivalued function, but we usually restrict its range to the interval $[0, \pi]$ to make it a single-valued function.

Q: How do you evaluate the expression $\operatorname{arcsec}(-\sqrt{2})$?

A: To evaluate the expression $\operatorname{arcsec}(-\sqrt{2})$, we need to find the angle whose secant is $-\sqrt{2}$. We can start by finding the angle whose secant is $\sqrt{2}$, which is $\frac{\pi}{4}$.

Q: What is the identity $\sec^2(x) = 1 + \tan^2(x)$ used for?

A: The identity $\sec^2(x) = 1 + \tan^2(x)$ is used to find the angle whose secant is $-\sqrt{2}$. We can use this identity to find the value of $\sec(x)$.

Q: How do you use the identity $\sec(x) = \frac{1}{\cos(x)}$ to find the angle whose secant is $-\sqrt{3}$?

A: We can use the identity $\sec(x) = \frac{1}{\cos(x)}$ to find the angle whose secant is $-\sqrt{3}$. We have:

$$-\sqrt{3} = \frac{1}{\cos(x)}$$

$$\cos(x) = -\frac{1}{\sqrt{3}}$$

Q: What is the identity $\cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x)$ used for?

A: The identity $\cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x)$ is used to find the angle whose cosine is $-\frac{1}{\sqrt{3}}$. We can use this identity to find the value of $x$.

Q: How do you use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$?

A: We can use the identity $\sin^{-1}(x) = \sin^{-1}\left(\frac{x}{|x|}\right)$ to find the angle whose sine is $\frac{1}{\sqrt{3}}$. We have:

$$x = \sin^{-1}\left(\frac{\frac{1}{\sqrt{3}}}{|\frac{1}{\sqrt{3}}|}\right)$$

$$x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Q: What is the final answer to the expression $\operatorname{arcsec}(-\sqrt{2})$?

A: The final answer to the expression $\operatorname{arcsec}(-\sqrt{2})$ is $\frac{3\pi}{4}$.

Q: Why is the inverse secant function important in mathematics?

A: The inverse secant function is important in mathematics because it is used to find the angle whose secant is a given value. It is also used in trigonometry and calculus to solve problems involving right triangles and circular functions.

Q: What are some common applications of the inverse secant function?

A: Some common applications of the inverse secant function include finding the angle of elevation or depression of a right triangle, finding the length of a side of a right triangle, and solving problems involving circular functions.

Q: How do you use the inverse secant function to solve problems involving right triangles?

A: To use the inverse secant function to solve problems involving right triangles, you need to find the angle whose secant is a given value. You can do this by using the identity $\sec(x) = \frac{1}{\cos(x)}$ and the inverse cosine function.

Q: What are some common mistakes to avoid when using the inverse secant function?

A: Some common mistakes to avoid when using the inverse secant function include:

• Not using the correct identity to find the angle whose secant is a given value.
• Not using the inverse cosine function to find the angle whose cosine is a given value.
• Not checking the domain and range of the inverse secant function.
• Not using the correct units when solving problems involving right triangles.

Q: How do you check the domain and range of the inverse secant function?

A: To check the domain and range of the inverse secant function, you need to make sure that the input value is within the domain of the function and that the output value is within the range of the function. The domain of the inverse secant function is all real numbers except for $-1$ and $1$, and the range is the interval $[0, \pi]$.