## Trigonometric Graphs

Note – The graphs of the Sine, Cosine and Tangent functions are considered higher topics on most exam boards.

The Sine Graph We can tell a great deal from the graph of the Sine function. Clearly the function is a bounded function between 1 and -1. This means that $\sin{x}$ will be no more than 1, and no less than -1 for any value of x. Feel free to check this on your calculator!

We can also see that the Sine function is an odd function. This means that, for example, $\sin{-x} = -\sin{x}$. This is clear for example, from the value of $90$; with $\sin{-90} = -1$ and $\sin{x} = 1$.

Furthermore, Sine is a periodic function, meaning that it repeats itself every set period. For the Sine function, this is every 360°. For any angle $x$, $\sin{(x)} = \sin{(x + 360)}$, (honestly, try it if you don’t believe me).

Did you know? – Because of the periodicity of the Sine function, it is commonly used to model phenomena such as sound and light waves, sunlight intensity and day length, and average temperature variations throughout the year.

The Cosine Graph At first sight, the graph of the Cosine function is very similar to that of the Sine function. In fact it’s actually the Sine graph, shifted to the right by 90°.

Similarly, the Cosine functions is a bounded function with limits -1 and 1, and also a periodic function with period of 360°. You can notice again that, for any angle $x$, $\cos{(x)} = \cos{(x + 360)}$

Unlike the Sine function however, the Cosine function is an even function. This means that $\cos{-x} = \cos{x}$ for any value $x$. Another way of visualising this is that an even function is symmetrical about the y-axis.

The Tangent Fuction The graph of the Tangent is the blue function on the graph above. The red lines are not part of the graph, but they are lines called asymptotes. These are used to show where the function “shoots off” to infinity.

The reason for this is that $\tan{x} = \frac{\sin{x}}{\cos{x}}$. Because you can’t divide by zero, $\tan{x}$ is undefined for values of $x$ such that $\cos{x} = 0$, (90°, 270°,…). It follows that $\tan{x} = 0$ for the same values that $\sin{x} = 0$.

The Tan function is also a periodic function with a period of 180°, however it is not bounded, unlike that of Sine and Cosine.