余弦函数的性质

Properties of Cosine

Cosine is a trigonometric function of an angle. It has a number of properties, such as being periodic and even. In the context of a triangle, for the specified angle, cosine is the ratio of the length of the side that is ADJACENT the angle to the length of the longest side, or HYPOTENUSE, of the triangle.

Once a triangle to analyze is chosen, we can write: $$\cos(\alpha) = \frac{adjacent}{hypotenuse}$$

Like I said above, the cosine function is periodic, which means that it is a function returning to the same value at regular intervals. Cosine has a period of $2\pi$, which means we can write it as: $$\cos(\alpha) = \cos(\alpha + 2\pi)$$ or, in a more general sense, $$\cos(\alpha) = \cos(\alpha + 2k\pi), k \in all \hspace{0.25cm} integers$$

As you can see from the graph above, the value of $\cos(\alpha)$ will always be between the values of $-1$ and $1$.

(Video) Properties of the Cosine Graph

by Math with Mister A

In this video by Math with Mister A, you will learn some of the main characteristics of cosine graphs.

Summary

Let's look at a graph of $y=cos(x)$.

The $y$-intercept

The $y$-intercept is $(0,1)$.

The $x$-intercepts

We see the first $x$-intercept at $\frac{\pi}{2}$ and a second $x$-intercept at $\frac{3\pi}{2}$. There is a distance of $\pi$ between the two intercepts. We can mathematically represent this as: $\frac{\pi}{2} + n\pi$, where $n$ is an integer.

The relative maximum

We see the first maximum at $0$, and a positive relative maximum at $2\pi$. On the negative side, we see a relative maximum at $-2\pi$. We can mathematically represent this as: $2\pi n$, where $n$ is an integer.

Function Directionality Over $(\frac{\pi}{2}, \frac{3\pi}{2})$

Looking at the graph, we will see the function both increasing and decreasing. So, over the given interval, it is neither increasing or decreasing.

The Cosine Function

The next type of trigonometric function we will learn about is the cosine function. This is the CAH part of SOHCAHTOA. The SOH is sine and the TOA is tangent.

Cosine functions follow the following formula:

$$\text{cos } {\theta} = \frac{\text{adjacent}}{\text{hypotenuse}}$$

The graph of the cosine function looks like:

Unlike the sine graph, the cosine graph starts at $y=1$ at the peak and $y=-2$ at the trough. The period is $2\pi$, and the amplitude is $1$. The cosine function crosses the $x$-axis at $x=\frac{\pi}{2}$ and $\frac{3\pi}{2}$. We can also use the unit circle to find cosine.

In later lessons, you will learn about the reciprocal of cosine, secant.