Trigonometry (HSC SSCE Mathematics Advanced): Revision Notes

Graphs of Trigonometric Functions

This revision note covers the graphs of the three main trigonometric functions: sine, cosine, and tangent. Understanding these graphs and their key features is essential for working with trigonometric functions in mathematics.

Sine and cosine functions

Connection to the unit circle

The sine and cosine functions arise naturally from the unit circle. When we look at a point on the unit circle, we can describe its position using trigonometry.

For a point on the unit circle at an angle $\theta$ measured anticlockwise from the positive $x$-axis, the coordinates of that point are $(\cos \theta, \sin \theta)$. This means the $x$-coordinate gives us the cosine value, and the $y$-coordinate gives us the sine value.

Values for common angles

It's helpful to know the sine and cosine values for commonly used angles. The table below shows sine values at key angles measured in radians:

$\theta$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\frac{5\pi}{6}$	$\pi$	$\frac{7\pi}{6}$	$\frac{4\pi}{3}$	$\frac{3\pi}{2}$	$\frac{5\pi}{3}$	$\frac{11\pi}{6}$	$2\pi$
$\sin \theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-1$	$-\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$0$

Similarly, cosine values follow a pattern around the unit circle. As the angle $\theta$ increases from $0$ to $2\pi$, the sine and cosine values move between $-1$ and $1$.

How the sine graph is created from the unit circle

The graph of $y = \sin \theta$ is derived directly from the unit circle. As we move around the circle through different values of $\theta$, we can track how the $y$-coordinate (which equals $\sin \theta$) changes.

This diagram shows how points on the unit circle correspond to points on the sine wave. The vertical dashed lines connect matching $\theta$ values between the circle and the graph. Notice how the sine value starts at $0$ when $\theta = 0$, rises to a maximum of $1$ at $\theta = \frac{\pi}{2}$, returns to $0$ at $\theta = \pi$, drops to a minimum of $-1$ at $\theta = \frac{3\pi}{2}$, and completes the cycle back at $0$ when $\theta = 2\pi$.

The complete sine function graph

Since sine is a circular function derived from the unit circle, it repeats its pattern indefinitely in both directions along the horizontal axis.

The graph shows the characteristic wave shape of the sine function. The smooth, continuous curve oscillates between $-1$ and $1$, crossing through zero at multiples of $\pi$.

How the cosine graph is created from the unit circle

Similarly, the cosine function can be visualized through the unit circle by tracking the $x$-coordinate as the angle changes.

The cosine graph has the same wave shape as the sine graph, but it starts at a different position. When $\theta = 0$, cosine equals $1$ (its maximum value), whereas sine equals $0$.

The complete cosine function graph

The graph of $y = \cos \theta$ extends infinitely in both directions, repeating every $2\pi$ units.

Both sine and cosine functions have similar properties and are described as circular functions or periodic functions. The term "circular" reflects their origin from the unit circle, while "periodic" describes how they repeat at regular intervals.

Key terminology: period

A crucial concept when studying trigonometric graphs is the period.

The period represents the length of one complete cycle in the graph. This is typically measured from a consistent reference point, such as from one peak to the next peak, or from one rising intercept to the next rising intercept.

For the sine and cosine functions, the period of $2\pi$ corresponds to one full revolution around the unit circle. After rotating through $2\pi$ radians, we return to the same position on the circle, so the sine and cosine values repeat.

Key terminology: amplitude and midline

In addition to period, the amplitude and midline are important features of trigonometric graphs.

Each graph oscillates around a horizontal line called the midline. For the basic sine and cosine functions ($y = \sin x$ and $y = \cos x$), the midline is the $x$-axis (the line $y = 0$), since the functions oscillate symmetrically between $-1$ and $1$.

In this diagram, we can clearly see the amplitude and period marked on the sine function. The amplitude of $1$ is the vertical distance from the midline ($y = 0$) to the maximum point. The period of $2\pi$ is the horizontal distance for one complete wave cycle.

For $y = \sin x$, the maximum value is $1$ and the minimum value is $-1$, so the amplitude is $\frac{1 - (-1)}{2} = \frac{2}{2} = 1$.

The cosine function has identical amplitude and period to the sine function. The only difference is the starting position of the wave.

Key features of y = sin x

Let's summarize all the important characteristics of the sine function:

Key features of y = cos x

Now let's examine the cosine function in the same way:

Tangent function

Definition of tangent

The tangent function is defined as the ratio of the sine function to the cosine function:

Understanding the period of tangent

The tangent function has an interesting property related to angles that differ by $\pi$. When we add $\pi$ to an angle, both the sine and cosine values change sign:

From the unit circle, we can see that:

• $\sin(\theta + \pi) = -\sin \theta$
• $\cos(\theta + \pi) = -\cos \theta$

The graph of y = tan θ

The tangent function has a very different appearance from sine and cosine.

Notice the distinctive features:

• The graph consists of separate curved branches
• Vertical dashed lines show positions where the function is undefined
• The pattern repeats every $\pi$ units
• The function values extend from negative infinity to positive infinity

Key features of the tangent function