Let $f(x)=1+3 \sec X$ And $g(x)=-5$. In The $xy$-plane, What Are The $x$-coordinates Of The Points Of Intersection Of The Graphs Of $f$ And $g$ For $0 \leq X \ \textless \ 2\pi$?A.

**Let $f(x)=1+3 \sec x$ and $g(x)=-5$. In the $xy$-plane, what are the $x$-coordinates of the points of intersection of the graphs of $f$ and $g$ for $0 \leq x \ \textless \ 2\pi$?**

Understanding the Problem

To find the points of intersection of the graphs of $f$ and $g$, we need to set the two functions equal to each other and solve for $x$. This is because the points of intersection occur where the two functions have the same $y$-value for a given $x$-value.

Setting Up the Equation

We are given the functions $f(x)=1+3 \sec x$ and $g(x)=-5$. To find the points of intersection, we set $f(x)=g(x)$ and solve for $x$.

$$1+3 \sec x = -5$$

Simplifying the Equation

To simplify the equation, we can start by isolating the $\sec x$ term.

$$3 \sec x = -6$$

Next, we can divide both sides of the equation by 3 to get:

$$\sec x = -2$$

Finding the Values of $x$

Now that we have the equation $\sec x = -2$, we can use the definition of the secant function to find the values of $x$ that satisfy this equation.

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

Substituting this into the equation, we get:

$$\frac{1}{\cos x} = -2$$

Taking the reciprocal of both sides, we get:

$$\cos x = -\frac{1}{2}$$

Solving for $x$

Now that we have the equation $\cos x = -\frac{1}{2}$, we can use the unit circle to find the values of $x$ that satisfy this equation.

The cosine function is negative in the second and third quadrants, so we need to find the angles in these quadrants that have a cosine value of $-\frac{1}{2}$.

Using the unit circle, we can see that the angles in the second and third quadrants that have a cosine value of $-\frac{1}{2}$ are:

$$x = \frac{2\pi}{3}, \frac{4\pi}{3}$$

Conclusion

Therefore, the $x$-coordinates of the points of intersection of the graphs of $f$ and $g$ for $0 \leq x \ \textless \ 2\pi$ are:

$$x = \frac{2\pi}{3}, \frac{4\pi}{3}$$

Q&A

Q: What is the definition of the secant function?

A: The secant function is defined as the reciprocal of the cosine function. It is denoted by $\sec x$ and is equal to $\frac{1}{\cos x}$.

Q: How do you find the values of $x$ that satisfy the equation $\sec x = -2$?

A: To find the values of $x$ that satisfy the equation $\sec x = -2$, we can use the definition of the secant function and the unit circle. We can see that the angles in the second and third quadrants that have a cosine value of $-\frac{1}{2}$ are the solutions to the equation.

Q: What are the $x$-coordinates of the points of intersection of the graphs of $f$ and $g$ for $0 \leq x \ \textless \ 2\pi$?

A: The $x$-coordinates of the points of intersection of the graphs of $f$ and $g$ for $0 \leq x \ \textless \ 2\pi$ are:

$$x = \frac{2\pi}{3}, \frac{4\pi}{3}$$

Q: How do you simplify the equation $1+3 \sec x = -5$?

A: To simplify the equation $1+3 \sec x = -5$, we can start by isolating the $\sec x$ term. We can then divide both sides of the equation by 3 to get $\sec x = -2$.