To solve $\text{arcsec}(2)$, we need to recall the definition and properties of the secant and inverse secant (arcsecant) functions.

Definition of arcsecant:

The arcsecant function, $\text{arcsec}(x)$, is the inverse of the secant function. This means that if: $\theta = \text{arcsec}(x)$ then: $\sec(\theta) = x$ and $\theta$ is typically restricted to the range $0 \leq \theta < \pi$, with $\theta \neq \frac{\pi}{2}$.

Solve $\text{arcsec}(2)$:

We are solving for $\theta$ such that: $\sec(\theta) = 2$ The secant is the reciprocal of the cosine: $\sec(\theta) = \frac{1}{\cos(\theta)}$ So we can rewrite the equation as: $\frac{1}{\cos(\theta)} = 2$ Now, solving for $\cos(\theta)$: $\cos(\theta) = \frac{1}{2}$

Step-by-Step Solution:

We need to find $\theta$ such that $\cos(\theta) = \frac{1}{2}$, and $\theta$ lies in the valid range for $\text{arcsec}(x)$.

• The cosine of $\theta = \frac{\pi}{3}$ is $\frac{1}{2}$.
• The secant function is positive in the first and fourth quadrants, but arcsec is defined in the first and second quadrants.

Thus, the correct value for $\theta$ is: $\theta = \frac{\pi}{3}$

Therefore: $\text{arcsec}(2) = \frac{\pi}{3}$

Would you like any further details on this, or a step-by-step breakdown?

Follow-up Questions:

1. What is the domain and range of the arcsecant function?
2. How does the secant function relate to the cosine function?
3. Can we solve $\text{arcsec}(-2)$? How?
4. What are some other important inverse trigonometric functions?
5. How do we solve for $\text{arccos}(x)$ and $\text{arcsin}(x)$?

Tip: Always check the domain restrictions when working with inverse trigonometric functions to avoid errors.