Graphs of Trigonometric Functions

Now that we have definitions of the sine and cosine of any real number x using the unit circle, we can now draw their graphs.

Sine and cosine graphs
In this demonstration, click on the START button. On the lefthandside you will see a ray being rotated in an anticlockwise direction. For each position of the ray, we generate an angle drawn in standard position. The angle is drawn in blue and the value of the sine of that angle is illustrated as the height of a green line. On the righthandside , we plot the size of the angle along the x-axis against the corresponding value of sin(x) along the y-axis. The curve generated is the graph of the sine function for values of x between 0 and 2.

By clicking on the SINE button, the corresponding situation for the cosine will be illustrated. The other buttons control the animation.

The illustration above only draws the graph of the sine or cosine function for one revolution. However, the domain of these functions is all real numbers. The picture below shows the graphs of sine (orange) and cosine (green) on a larger domain (-3 to 3).

Observe that the values for both functions repeat every 2. This was an observation we made earlier about the sine and cosine of coterminal angles. Functions whose values repeat on an interval of length p are called periodic of period p. Thus the sine and cosine are periodic of period 2. Note that the range of values of both functions is -1 to 1 inclusive.

Another thing to notice is the symmetry of the sine and cosine functions. The sine function is an odd function. Recall this means that if (x,y) is on the graph of the function so too is the point (-x,-y). Since y corresponds to sin(x) then this means that sin(-x) = - sin(x).
The cosine is an even function which means that if (x,y) is on the graph of the function so too is the point (-x,y). Since y corresponds to cos(x) then this means that cos(-x) = cos(x).

These symmetries give rise to the identities summarized in the following table.

sin(-x) = -sin(x)
cos(-x) = cos(x)

We also note that the sine function has value 0 at the values n, n an integer. The cosine function takes on zero value at /2 + n, n an integer. This is very important since the other four trigonometric functions involve reciprocals of the sine and cosine functions. For example, tan(x) = sin(x)/cos(x) and so the tangent function is undefined at /2 + n, n an integer. We summarize these results.

FunctionUndefined at
sine
cosine
-
secant
tangent
/2 + n, n an integer.
cosecant
cotangent
n, n an integer.

Indeed, each of the remaining four functions has a vertical asymptote at each real number where they are undefined. Here are the graphs of the remaining four trigonometric functions. Each is drawn over the interval -3 to 3.

tan(x) cot(x)
sec(x) csc(x)

Unlike sine and cosine, the graphs of tangent and cotangent repeat on an interval of length . That is, the tangent and cotangent are periodic of period . This gives us the following identities:

tan(x+) = tan(x)
cot(x+) = cot(x)

From the graphs above, we see that tan, cot and csc are odd functions while sec is an even function. These symmetries give us the following identities.

tan(-x) = - tan(x)
cot(-x) = - cot(x)
csc(-x) = - csc(x)
sec(-x) = sec(x)

These identities may be obtained directly from the symmetries of the sine and cosine functions. For example, tan(-x) = sin(-x)/cos(-x) = - sin(x)/cos(x) = -tan(x). Similarly, sec(-x) = 1/cos(-x) = 1/cos(x) = sec(x).