Trigonometric Graphs

students, have you ever noticed how a ferris wheel goes up and down in a smooth pattern 🎡, or how ocean waves repeat over time 🌊? Trigonometric graphs are the math tools that describe those repeating patterns. In this lesson, you will learn what the graphs of $\sin x$, $\cos x$, and $\tan x$ look like, how to read their features, and how to use them in real situations. By the end, you should be able to explain the key terms, sketch the graphs, and connect them to applied problems in geometry and trigonometry.

What a trigonometric graph shows

A trigonometric graph is a picture of how a trigonometric function changes as the input changes. The input is usually an angle, measured in radians or degrees. The output tells you the value of the function. For example, in the graph of $y=\sin x$, the $x$-axis shows the angle, and the $y$-axis shows the sine value.

The most important idea is that trigonometric graphs are periodic. That means the pattern repeats after a fixed amount of input. For $y=\sin x$ and $y=\cos x$, the period is $2\pi$. For $y=\tan x$, the period is $\pi$.

A graph also helps you see the following features:

Amplitude: the maximum distance from the midline to a peak or trough.
Midline: the horizontal line halfway between the maximum and minimum values.
Period: the length of one complete cycle.
Phase shift: a horizontal shift left or right.
Vertical shift: a move up or down.

These features are very useful because they let you model real-world repeating situations, such as seasons, tides, sound waves, and rotating machinery ⚙️.

The graph of $y=\sin x$

The graph of $y=\sin x$ is one of the most important graphs in mathematics. It starts at the point $(0,0)$, rises to a maximum of $1$, returns through $0$, drops to a minimum of $-1$, and then repeats.

Key points on one cycle are:

$(0,0)$
$\left(\frac{\pi}{2},1\right)$
$(\pi,0)$
$\left(\frac{3\pi}{2},-1\right)$
$(2\pi,0)$

This graph is smooth and continuous. Its amplitude is $1$, its midline is $y=0$, and its period is $2\pi$.

Why is this graph useful? Imagine a point moving around a circle at a constant speed. The vertical position of that point changes like a sine wave. This is why sine is often used to model oscillations, such as a child on a swing or a speaker cone moving in and out 🔁.

If the function is $y=a\sin x$, then the amplitude becomes $|a|$. For example, $y=3\sin x$ has amplitude $3$, so the highest and lowest values are $3$ and $-3$.

The graph of $y=\cos x$

The graph of $y=\cos x$ is very similar to the sine graph, but it starts at its maximum value instead of at $0$. The basic cosine graph begins at $(0,1)$.

Important points on one cycle are:

$(0,1)$
$\left(\frac{\pi}{2},0\right)$
$(\pi,-1)$
$\left(\frac{3\pi}{2},0\right)$
$(2\pi,1)$

The cosine graph has the same amplitude, midline, and period as sine: amplitude $1$, midline $y=0$, and period $2\pi$.

Sine and cosine are closely related. In fact, $\cos x$ is just a shifted version of $\sin x$ because $\cos x=\sin\left(x+\frac{\pi}{2}\right)$. This means the two graphs have the same shape but different starting points.

A real-world example is daylight over the year. In many places, the amount of daylight changes in a repeating pattern. A cosine graph can model this because it begins at a maximum or minimum, depending on how the situation is set up ☀️.

The graph of $y=\tan x$

The tangent graph behaves differently from sine and cosine. Since $\tan x=\frac{\sin x}{\cos x}$, it is undefined whenever $\cos x=0$.

The basic tangent graph:

passes through $(0,0)$
increases from left to right
has vertical asymptotes at $x=\frac{\pi}{2}+k\pi$, where $k\in\mathbb{Z}$
repeats every $\pi$

A vertical asymptote is a line that the graph gets closer to but never touches. For tangent, the graph grows without bound near the asymptotes.

The tangent graph is useful in problems involving steepness, slope, and angles of elevation. For example, if you know the height of a building and the distance from it, you can use tangent to find the angle of elevation. In graph form, this helps you understand how the ratio changes as the angle changes.

Because tangent is undefined at certain values, its graph has gaps. That is a major difference from the smooth looping shapes of sine and cosine.

Transformations of trigonometric graphs

Many exam questions use transformed trigonometric graphs. A general sine or cosine function can be written as

$$y=a\sin\bigl(b(x-c)\bigr)+d$$

$$y=a\cos\bigl(b(x-c)\bigr)+d.$$

Here is what each part means:

$a$ affects amplitude and reflection in the $x$-axis.
$b$ affects the period.
$c$ gives the phase shift.
$d$ gives the vertical shift.

The period becomes

$$\frac{2\pi}{|b|}$$

for sine and cosine.

For example, for $y=2\sin\bigl(3(x-\frac{\pi}{6})\bigr)-1$, the amplitude is $2$, the period is $\frac{2\pi}{3}$, the graph shifts right by $\frac{\pi}{6}$, and the midline is $y=-1$.

This is exactly the kind of reasoning used in IB Mathematics: Applications and Interpretation HL. You are not only identifying a shape, but also interpreting the meaning of each parameter in a function.

A negative value of $a$ reflects the graph in the midline. For example, $y=-\sin x$ is the sine graph flipped upside down.

Reading and sketching trigonometric graphs

When sketching a graph, start by finding the important features. For sine and cosine, identify:

Amplitude
Period
Midline
Phase shift
Key points over one cycle

Suppose you need to sketch $y=4\cos\left(\frac{x}{2}\right)$. First, compare it to $y=a\cos(bx)$. Here, $a=4$ and $b=\frac{1}{2}$. So the amplitude is $4$ and the period is

$$\frac{2\pi}{\frac{1}{2}}=4\pi.$$

The graph begins at $(0,4)$ because cosine starts at its maximum when there is no phase shift. It reaches its minimum at $x=2\pi$, crosses the midline at $x=\pi$ and $x=3\pi$, and returns to the maximum at $x=4\pi$.

A good sketch does not need every point, but it should show the correct shape and important features. In IB exams, marks are often awarded for correct method, correct key values, and clear reasoning.

Why trigonometric graphs matter in Geometry and Trigonometry

Trigonometric graphs connect strongly to the rest of Geometry and Trigonometry because they turn angle relationships into visual patterns. They help you move between triangles, circles, and motion.

For example:

In right triangles, trigonometric ratios such as $\sin \theta$, $\cos \theta$, and $\tan \theta$ connect an angle to side lengths.
On the unit circle, the coordinates $(\cos \theta,\sin \theta)$ show how angles create graph values.
In applied settings, graphs can model height, distance, rotation, and cyclic change.

This is useful in three-dimensional reasoning too. A rotating object in space may produce a repeating projection on a line or plane. The graph helps represent that motion over time.

A simple applied example is a tide model. If the water level rises and falls regularly, a trigonometric graph can estimate the height at any time $t$. If the average water level is $5$ m, the amplitude is $2$ m, and one full cycle takes $12$ hours, a suitable model could be

$$h(t)=2\sin\left(\frac{\pi}{6}t\right)+5,$$

where $h(t)$ is the height in meters and $t$ is measured in hours. This model tells you the water level at different times and shows the repeating pattern clearly 🌊.

Conclusion

Trigonometric graphs are a central part of Geometry and Trigonometry because they describe repeated change in a clear visual way. students, you should now be able to identify the main graphs of $y=\sin x$, $y=\cos x$, and $y=\tan x$, explain key terms such as amplitude, period, midline, and phase shift, and use transformations to interpret and sketch more complex graphs. These ideas are important not only for exam questions, but also for modeling real-life situations where patterns repeat.

Study Notes

$\sin x$ and $\cos x$ are periodic with period $2\pi$.
$\tan x$ is periodic with period $\pi$ and has vertical asymptotes where $\cos x=0$.
Amplitude is the distance from the midline to a maximum or minimum value.
The midline of $y=a\sin\bigl(b(x-c)\bigr)+d$ or $y=a\cos\bigl(b(x-c)\bigr)+d$ is $y=d$.
The period of sine and cosine in transformed form is $\frac{2\pi}{|b|}$.
A phase shift is controlled by $c$ in $x-c$.
A negative value of $a$ reflects the graph in the midline.
Sine and cosine graphs are the same shape but shifted horizontally.
Trigonometric graphs are useful for modeling waves, tides, sound, rotation, and seasonal change.
In IB problem solving, always identify the graph’s key features before sketching or interpreting it.