What are the key differences between the graphs of $\tan(x)$ and $\sin(x)$ ?

Tangent: Vertical asymptotes, period of $\pi$ , range of $(-\infty, \infty)$ . Sine: No asymptotes, period of $2\pi$ , range of $[-1, 1]$ .

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What are the key differences between the graphs of $\tan(x)$ and $\sin(x)$ ?

Tangent: Vertical asymptotes, period of $\pi$ , range of $(-\infty, \infty)$ . Sine: No asymptotes, period of $2\pi$ , range of $[-1, 1]$ .

Compare the period of $\tan(x)$ and $\cos(x)$ .

Tangent: Period is $\pi$ . Cosine: Period is $2\pi$ .

Compare the range of $\tan(x)$ and $\sin(x)$ .

Tangent: Range is $(-\infty, \infty)$ . Sine: Range is $[-1, 1]$ .

What are the key differences between transformations of $\tan(x)$ and $\sin(x)$ ?

Period changes are calculated differently. Tangent: $T = \frac{\pi}{|b|}$ . Sine: $T = \frac{2\pi}{|b|}$ . Asymptotes exist for tangent but not for sine.

Compare the asymptotes of $\tan(x)$ and $\cot(x)$ .

Tangent: Asymptotes at $x = \frac{\pi}{2} + k\pi$ . Cotangent: Asymptotes at $x = k\pi$ (where k is an integer).

Compare the behavior of $\tan(x)$ and $\sin(x)$ near $x = 0$ .

As $x$ approaches 0, $\tan(x)$ approaches 0. As $x$ approaches 0, $\sin(x)$ approaches 0.

Compare the symmetry of $\tan(x)$ and $\cos(x)$ .

Tangent: Odd function, symmetric about the origin. Cosine: Even function, symmetric about the y-axis.

Compare the domain of $\tan(x)$ and $\sin(x)$ .

Tangent: All real numbers except $x = \frac{\pi}{2} + k\pi$ . Sine: All real numbers.

Compare the graphs of $y = \tan(x)$ and $y = \cot(x)$ .

Tangent: Increasing between asymptotes. Cotangent: Decreasing between asymptotes. Asymptotes are located in different places.

Compare the effect of 'a' in $y = a\tan(x)$ and $y = a\sin(x)$ .

Tangent: Vertical stretch/compression, reflection if 'a' is negative. Sine: Amplitude change, reflection if 'a' is negative.

What does the steepness of the tangent function's graph indicate?

The steepness indicates the rate of change of the function. Near the asymptotes, the function changes very rapidly.

How can you identify the period of a tangent function from its graph?

The period is the distance between two consecutive vertical asymptotes.

How does the graph of $y = -\tan(x)$ differ from $y = \tan(x)$ ?

The graph of $y = -\tan(x)$ is a reflection of $y = \tan(x)$ over the x-axis.

How can you identify a phase shift from the graph of a tangent function?

Compare the location of the asymptotes to the standard $y = \tan(x)$ graph. A horizontal shift in the asymptotes indicates a phase shift.

What does a vertical shift do to the graph of a tangent function?

It moves the entire graph up or down, changing the y-coordinates of all points on the graph.

How does the graph of $y = a \tan(x)$ change when $|a| > 1$ ?

The graph is vertically stretched, making it steeper compared to the graph of $y = \tan(x)$ .

How does the graph of $y = \tan(bx)$ change when $b > 1$ ?

The graph is horizontally compressed, decreasing the period and bringing the asymptotes closer together.

What does the symmetry of the tangent function about the origin indicate?

It indicates that the tangent function is an odd function, meaning $\tan(-x) = -\tan(x)$ .

If a tangent graph has asymptotes at $x = 0$ and $x = \pi$ , what is its period?

The period is $\pi$ (the distance between the asymptotes).

How can you tell from a graph if a tangent function has been reflected across the x-axis?

The function will decrease from left to right between asymptotes, instead of increasing.

What is the formula for $\tan(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$ ?

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

What is the general equation for a transformed tangent function?

$y = a \tan(b(x - c)) + d$

What is the formula for the period $T$ of a tangent function?

$T = \frac{\pi}{|b|}$

How do you calculate the location of vertical asymptotes for $y = \tan(x)$ ?

$x = \frac{\pi}{2} + k\pi$ , where $k$ is an integer.

How does 'a' affect the tangent function $y = a \tan(x)$ ?

'a' controls the vertical dilation. If 'a' is negative, the function is reflected over the x-axis.

How does 'b' affect the tangent function $y = \tan(bx)$ ?

'b' affects the period of the function. The period is $T = \frac{\pi}{b}$ .

How does 'c' affect the tangent function $y = \tan(x-c)$ ?

'c' shifts the graph horizontally. Positive 'c' shifts right, negative 'c' shifts left.

How does 'd' affect the tangent function $y = \tan(x)+d$ ?

'd' shifts the graph vertically. Positive 'd' shifts up, negative 'd' shifts down.

What is $x$ on the unit circle?

$x = cos(\theta)$

What is $y$ on the unit circle?

$y = sin(\theta)$