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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 $y = \tan(x)$.

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.

Figure 2.7.1. (left) $\theta$ in relation to $\tan(\theta)$, geometrically. (right) Graph of $y=\tan(x)$ for $0\le x\le 2\pi$.

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 $\sec(x) = \dfrac{1}{\cos(x)}$.
  2. The cosecant function, csc, is defined as $\csc(x) = \dfrac{1}{\sin(x)}$.
  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.

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