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# 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

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. 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.