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