Properties of trigonometric functions

The following graphs show the parent functions for $\sin$, $\cos$, and $\tan$. Sine and cosine graphs are often called waves because of their repeating pattern. You’ll study the details of these wave-like characteristics more later.

Sine

A sine curve looks like a wave, so it has wave-like properties such as amplitude and period.

The amplitude of a wave is the vertical distance from the $x$-axis (the midline) to a peak. The distance is the same to a peak above the axis and to a peak below the axis. For the parent function $y=\sin (x)$, the amplitude is $1$.

Amplitude tells you the maximum and minimum values of the graph. Since the amplitude of $y=\sin (x)$ is $1$, its maximum value is $1$ and its minimum value is $-1$.

The period of a wave is the horizontal distance needed to complete one full cycle of the pattern. For a sine graph, a helpful way to remember one cycle is:

“middle-high-middle-low-middle.”

This means the curve starts on the $x$-axis, rises to $y=1$, returns to the $x$-axis, drops to $y=-1$, and returns to the $x$-axis.

In the example above, one period begins at $x=0$ and ends at $x=2\pi$, so the period is $2\pi$. The halfway point of the period is at $x=\pi$.

Cosine

The parent cosine graph has the same amplitude as the parent sine graph: $1$. That means the maximum value of $y=\cos (x)$ is $1$ and the minimum value is $-1$.

The period is also $2\pi$. The main difference is where the cycle starts. A cosine cycle can be remembered as:

“high-middle-low-middle-high.”

It begins at $y=1$ when $x=0$, decreases to $y=-1$, then increases back to $y=1$.

Tangent

The graph of $y=\tan (x)$ looks very different from sine and cosine. It isn’t wave-shaped. Instead, it consists of repeating curves that look somewhat like pieces of a cubic function.

Notice the vertical asymptotes that separate each curve. If you need a refresher on asymptotes, review the chapter Nonlinear functions and graphs.

Starting at the origin, the curves intersect the $x$-axis at intervals of $\pi$ in both the positive and negative $x$ directions. The vertical asymptotes also occur at intervals of $\pi$ in both directions, but they start at $x=\pi /2$.

Nonparent functions

You’ll often see sine and cosine functions that differ from their parent functions. In those cases, you need to find the amplitude and period from the equation (without relying on a graph).

You can find the amplitude of a sine or cosine wave by looking at the number multiplied in front of the function.

• In $y=\sin (x)$, there is no visible number in front, so the coefficient is $1$. That’s why the amplitude of the parent sine and cosine functions is $1$.

Here is another example:

Find the amplitude of the following function:

$$y=3\ast \sin (2x)$$

The amplitude of the function is $3$. This is also the maximum value of the function. The minimum value is $-3$.

To find the period, figure out what value you need to multiply by so that the coefficient of $x$ inside the function becomes $2\pi$. For the parent functions, the inside is just $x$, so the coefficient is $1$, and you multiply by $2\pi$ to get $2\pi$.

Here is an example of finding the period:

Find the period of the following function:

$$y=3\ast \sin (2x)$$

Find what value you need to multiply by so that the coefficient of $x$ becomes $2\pi$. The current coefficient is $2$.

$$2\ast \text{period}=2\pi$$

The period of the function is $\pi$.

Trigonometric identity

Trigonometric identities are equations that show relationships between trigonometric functions. For the ACT, you only need to know this one:

$${\sin }^2(x)+{\cos }^2(x)=1$$

The $x$ in this identity can be any expression, as long as it is exactly the same in both functions. For example:

$${\sin }^2(2x^2)+{\cos }^2(2x^2)=1$$