If you already know the relationship between the equation and graph of sine and cosine functions then the other four functions can be found by identifying zeroes, asymptotes and key points. Are the four new functions transformations of the sine and cosine functions?
Graphing Other Trigonometric Functions
Secant and Cosecant
Since secant is the inverse of cosine the graphs are very closely related.
Notice wherever cosine is zero, secant has a vertical asymptote and where
The method is to graph it as you would a cosine and then insert asymptotes and the secant curves so they touch the cosine curve at its maximum and minimum values. This technique is identical to graphing cosecant graphs. Simply use the sine graph to find the location and asymptotes.
Tangent and Cotangent
The tangent and cotangent graphs are more difficult because they are a ratio of the sine and cosine functions.
The way to think through the graph of
By plotting all this information, you get a very good sense as to what the graph of tangent looks like and you can fill in the rest.
Notice that the period of tangent is
The graph of cotangent can be found using identical logic as tangent. You know
Earlier, you were asked if the four new functions are transformations of sine and cosine. The four new functions are not purely transformations of the sine and cosine functions. However, secant and cosecant are transformations of each other as are tangent and cotangent.
Graph the function
Graph the function as if it were a sine function. Then insert asymptotes wherever the sine function crosses the sinusoidal axis. Lastly add in the cosecant curves.
The amplitude is 2. The shape is negative sine. The function is shifted up one unit and to the right one unit.
Note that only the blue portion of the graph represents the given function.
How do you write a tangent function as a cotangent function?
There are two main ways to go between a tangent function and a cotangent function. The first method was discussed in Example A:
The second approach involves two transformations. Start by reflecting across the
Find the equation of the function in the following graph.
If you connect the relative maximums and minimums of the function, it produces a shifted cosine curve that is easier to work with.
The amplitude is 3. The vertical shift is 2 down. The period is 4 which implies that
Where are the asymptotes for tangent and why do they occur?
1. What function can you use to help you make a sketch of
2. What function can you use to help you make a sketch of
Make a sketch of each of the following from memory.
Graph each of the following.
14. Find two ways to write
15. Find two ways to write
To see the Review answers, open this PDF file and look for section 5.7.