We can now state the meanings 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 magnitude of the slope of the line through the circle’s center that forms an angle of x radians.

The animation in Figure 2.7.1 shows the value of tan(x) in relation to the value of x, where x is in radians. Study Figure 2.7.1 in relation to Figure 2.6.2. Make sure that you see the same thing happening in both.

Figure 2.7.1. Graph of y = tan(x) for x ≥ 0.

Reflection 2.9. The vertical axis in Figure 2.16 is labeled “y
(radians)”. Is that appropriate, or should it be labeled “y (radii)”?

Reflection 2.10. What do you think the faint vertical lines mean in
the graph of $y = \tan(x)$?

Reflection 2.11. Does tan(x) ever decrease? What does it mean
for a function to decrease?

Reflection 2.12. At what values of x does the graph of $y = \tan(x)$ “break”? Remember, values of x are in radians.

Exercise Set 2.7

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. 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.

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.

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.