正切函数的性质

The Tangent Function

Tangent is the ratio between sine and cosine.

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

The sine function is the numerator, while the cosine function is the denominator. To find the tangent, you will use this ratio.

We can also think of the tangent as the ratio between the opposite side of an angle and the adjacent side of the angle on a triangle: $\frac{opposite}{adjacent}$.

You may remember learning about tangent when you learned about the unit circle.

Graphically:

You'll notice that this function is a periodic function. Here are some key properties:

• These are odd functions because $tan(-x) = -tan(x)$.

• If $cos(x)=0$, then the function is undefined, as the denominator of a fraction cannot equal $0$.

• There are an infinite number of vertical asymptotes.

Here are some important tangent values to memorize:

• $tan(0) = 0$

• $tan(\frac{\pi}{6}) = \frac{1}{\sqrt3}$

• $tan(\frac{\pi}{4}) = 1$

• $tan(\frac{\pi}{3}) = \sqrt3$

• $tan(\frac{\pi}{2}) = \text{undefined}$

The Tangent Function

Tangent is the TOA of SOHCAHTOA.

We can think of tangent as the ratio between sine and cosine.

$$\text{ tan }(x) = \frac{\text{ sin }(x)}{\text{ cos }(x)}$$

We can also think of tangent as the ratio between the length of the opposite side and the length of the adjacent side.

$$\text{ tan }(x) = \frac{\text{ opposite }}{\text{ adjacent }}$$

The graph looks like:

The period of the function is $\pi$. The next lesson will cover some of the important features of graphing tangent functions.

In later lessons, you will learn about the reciprocal of tangent, cotangent. You will also learn how to use the unit circle to find tangent.

(Video) The Tangent Function

by Mister TamGoesHam

In this video by Mister TamGoesHam, you will learn about the tangent function.

Summary

You may remember that the tangent function tells us that:

$$\text{ tan }=\frac{\text{opposite}}{\text{adjacent}} = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} = \frac{y}{x}$$

When it comes to our unit circle, we will use the above formula to evaluate. If we are looking for the tangent at the point $(0,1)$, we will evaluate at $\text{ tan }(\theta)=\frac{0}{1} = 0$. We can do the same thing for any other point around the circle. If we want to evaluate the tangent of $\frac{\pi}{6}$, we will look at our unit circle and say that: $\text{ tan }(\theta)=\frac{\frac{1}{2}}{\frac{\sqrt3}{2}} = \frac{\sqrt3}{3}$. Follow along with the video to continue finding points along the unit circle.

We can plot these points in a more familiar setting - the coordinate plane! From $-\frac{\pi}{2}$ to $\frac{\pi}{2}$, our tangent function graph looks like:

Some key notes:

• The vertical asymptotes is: $x=-\frac{\pi}{2}$ and $x=\frac{\pi}{2}$

• The point of inflection is at the origin, $(0,0)$

• The period is $\pi$, which will be important when we are transforming the function