5.4 Video 1 of 2:More Trig Graphs

5.4 Video 2 of 2:Even More Trig Graphs

Section Notes

More Trigonometric Graphs

While the last section focused on the graphs of Sine and Cosine, this section will focus on the graphs of the other trigonometric functions. These functions also have periodic properties. and are and are only periodic.

Periodic Properties of

When graphing a period of a tangent function, we are forced to deal with the fact that Tangent is undefined at any odd multiple of . Calculating values of tangent near it can be seen that the tangent values increase sharply as the values get closer and closer to (but do not exceed) . This is evident on the graph as a vertical asymptote:

We can also see in the calculations of cotangent, that the values have an asymptote at 0 and any multiple of

And graphing multiple periods of each function yields:

The cosecant and secant graphs can be visualized by the fact that they are the multiplicative reciprocals of sine and cosine respectively. Thus, the graph of should appear to be multiplicative inverse of . One period of the cosecant function is graphed below:

And extending this for multiple periods gives the periodic graph:

And likewise, the graph of can be generated as is the multiplicative inverse of :

These trigonometric functions can be shifted, similarly to the shifts we saw in the cosine and sine graphs. Here are a few vertical stretches of the tangent function on a single graph:

Note that the vertical stretch does not alter the period of the tangent function. A tangent function can have its period altered, however, by multiplying the variable:

and the tangent graph can be shifted vertically and horizontally:

We can formalize these changes for tangent and cotangent curves:

Tangent and Cotangent Graphs
The tangent and cotangent curves Are stretched vertically by a factor of , have a period of are horizontally shifted by , and vertically shifted by . One period is graphed between the interval

The graphs of Secant and Cosecant are similarly manipulated:

Secant and Cosecant Graphs
The secant and cosecant curves Are stretched vertically by a factor of , have a period of are horizontally shifted by , and vertically shifted by . One period is graphed between the interval