Sine Function Properties

Properties of Sine

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

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

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

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

We also mentioned that the sine function is an odd function, which means that the sine of a negative angle equals the negative sine of an angle, or: $$\sin(-\alpha) = - \sin(\alpha)$$

(Video) Properties of the Sine Graph

by Math with Mister A

In this video by Math by Mister A, you will learn about some of the major characteristics of sine graphs.

Summary

Here is the a graph of the sine function:

The range of the function is from $[-1,1]$. The maximum is at $y=1$, and the minimum is at $y=-1$, with points continuously between these two points. The $x$-intercepts are at $0$, $\pi$, $2\pi$, $3\pi$, etc., on the positive side. On the negative side, the $x$-intercepts are at $-\pi$, $-2\pi$, etc. We can represent the $x$-intercepts as $n\pi$, where $n$ is an integer.

The relative maximum is at $\frac{\pi}{2}$ and $\frac{5\pi}{2}$. We need to go a distance of $2\pi$ units to get from one relative maximum to another, so we can mathematically represent the relative maximums with: $\frac{\pi}{2} + 2\pi n$, where $n$ is an integer.

The relative minimum is at $\frac{3\pi}{2}$ and $\frac{-\pi}{2}$. We need to go a distance of $2\pi$ units to get from one relative minimum to another, so we can mathematically represent the relative minimums with: $\frac{3\pi}{2} + 2\pi n$, where $n$ is an integer.

Properties of the Sine Function

Let's look at the SOH in SOHCAHTOA! There are a few key characteristics of sine.

We know from SOHCAHTOA, that sine functions are: $$sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$

Here is the graph:

The graph of $y=sin(\theta)$ starts at $y=0$ then peaks at $y=1$. The curve then goes down, crosses the $x$-axis at $x=\pi$, and hits the trough at $y=1$. The graph then goes back up to touch the $x$-axis at $x=2\pi$. Sine is a periodic function, so this cycle repeats (think: the unit circle!).

We can also use the unit circle to find sine.

You can also do transformations. One example is a phase shift, where the function starts at a value on the $x$-axis that is not at the origin. You can think of a phase shift as a horizontal translation.

In later lessons, you will learn about the inverse of the sine, arcsine. You will also learn about the reciprocal of sine, cosecant.