The trigonometric graphs of sin, cos and tan can be derived from the unit circle. Imagine a particle going round the unit circle. Let $\theta$ measure the anti-clockwise angle between the particle and the x-axis. The value of $\cos(\theta)$ can be thought of as the x coordinate of the particle. Similarly, the value of $\sin(\theta)$ can be thought of as the y coordinate. See the table  below.  You may be expected to memorise these values. Additionally, you should also know values of sin and cos at 30 and 60 degrees. See Trigonometric Identities.

## The graphs of cos and sin

We now plot the x-coordinate of the particle against the angle. Calculating the coordinates for all $\theta$ using a scientific calculator will give the following curve: Local lines of symmetry are vertical lines at every 180 degrees. In other words, they have equation $x=180n$ where $n\in{\mathbb Z}$ (what does this mean?).

We can also plot the y-coordinate against the angle of the particle. Calculating the coordinates for all $\theta$ using a scientific calculator will give the following curve: Local lines of symmetry are vertical lines every 180 degrees starting at 90 degrees. In other words, they have equation $x=180n-90$ (or equivalent) where $n\in{\mathbb Z}$.

Both y values on the sin and cos graphs are bounded between -1 and 1. The tan graph, however, can take values between $-\infty$ and $\infty$. Both cos and sin graphs repeat every 360 degrees. We say that they have periodicity of 360 degrees.
When solving trigonometric equations, scientific calculators tend to only give default solutions. However, the lines of symmetry above can be used to find all solutions in a given interval. See Trigonometric Equations.

For obvious reasons, sin and cos graphs are sometimes called waves. You may also be interested in the amplitude and wavelength. Firstly, amplitude is the height of the wave. Secondly, the wavelength is how long a single wave is.

## The graph of tan

Recall that $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$ from the trigonometric identities. The graph of $\tan(\theta)$ is given below. It can be thought of as the ratio of vertical to horizontal components of the above particle. As mentioned above, tan is not restricted to taking values between -1 and . Tan can in fact have any positive or negative value. However, the tan graph has asymptotes – lines that are never touched. These lines occur every 180 degrees starting at 90. They appear as a result of dividing by $\cos(\theta)=0$ in the expression $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.  Tan has no reflectional lines of symmetry but it does have a periodicity of 180 degrees.

## Examples – Transformations to Trigonometric Graphs

You may be expected to perform transformations to trigonometric graphs. See Examples 1 and 2 here. Click here for a reminder of Transformations.

### Example 1

Sketch the graph of $y=2\cos(\theta)$. First notice that this is a y-transformation. Namely, the y coordinates have been multiplied by 2. This stretches the graph by a factor of 2 in the y direction. See more on transformations.

### Example 2

Sketch the graph of $y=\sin(2\theta)$. Notice now that this is an x-transformation. That is, the x coordinates have been multiplied by 2. This stretches the graph by a factor of a half in the x direction. See more on transformations.

### Example 3 – C2 Jan 2012 Question 9ii Note that this question requires knowledge of radians (coming soon). This is an alternative angle measurement to degrees. Specifically, 180 degrees is $\pi$ radians. This means that $360^\circ=2\pi$ etc.

You will also need to be very careful with the transformations. Namely, identify the transformations when replacing x with ax-b carefully.  Firstly, replace x with x-b. Then replace new x with ax. Of course that is if you are choosing to use transformations. An alternative would be to substitute coordinates.

See Past Papers.

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