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.