|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|
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).|
|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.
|/2 + n, n an integer.|
|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.
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).