Section 2.7
The Meaning of x and y in y = tan(x)

• x is an arc length measured in units of a circle’s radius, rolled onto the x-axis.
• y is the slope of the line through the circle’s center that forms an angle of x radians.

Figure 2.6.2 is repeated below, on the left side of Figure 2.7.1. It shows $\tan(\theta)$ in relation to $\theta$.

The animation on the right of Figure 2.7.1 shows the value of $\tan(x)$ in relation to the value of x in the x-y plane, where x is in radians.

Study the animations in Figure 2.7.1 until you see the same thing happening in both animations.

Reflection 2.7.1. The vertical axis in Figure 2.7.1 (right) is labeled "y (radians)". Is that appropriate, or should it be labeled "y (radii)"?

Reflection 2.7.2. What do you think the faint vertical lines mean in the graph of $y = \tan(x)$?

Reflection 2.7.3. Does tan(x) ever decrease? What does it mean for a function to decrease?

Reflection 2.7.4. At what values of x does the graph of $y = \tan(x)$ "break"? Remember, values of x are in radians.

Exercise Set 2.7

1. The secant function, sec, is defined as $\sec(x) = \dfrac{1}{\cos(x)}$.
   • Type $y = \cos(x)$ into GC.
   • On a second line, type $y’ = \sec(x’)$ into GC. (The prime notation tells GC to use a second graphing pane.)
   • Press Enter. Widen GC's window and recenter the graphs if necessary.

   Use the graph of $y = \cos(x)$ and the definition of the sec function to explain why the graph of $y = \sec(x)$ behaves as it does.

2. The cosecant function, csc, is defined as $\csc(x) = \dfrac{1}{\sin(x)}$.
   • Type $y = \sin(x)$ into GC.
   • On a second line, type $y’ = \csc(x’)$ into GC. (The prime notation tells GC to use a second graphing pane.)
   • Press Enter.

   Use the graph of $y = \sin(x)$ and the definition of the csc function to explain why the graph of $y = \csc(x)$ behaves as it does.

3. The function tan is defined as tan(x) = $\dfrac{\sin(x)}{\cos(x)}$. The cotangent function is defined as $\cot(x) = \dfrac{1}{\tan(x)}$.
   
   Display a graph $y = \tan(x)$. Use it and the definition of the cotangent function to explain why the graph of $y = \cot(x)$ behaves as it does.