Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
In this video lesson, we talk about the graphs of the three other trigonometric functions of cosecant, secant, and cotangent.
Remember our basic three trigonometric functions are sine, cosine, and tangent. The three functions that we are talking about today are defined as the reciprocals of our basic three functions. So, we have cosecant (csc) is the reciprocal of sine or 1/sine, secant (sec) is the reciprocal of cosine or 1/cosine, and cotangent (cot) is the reciprocal of the tangent function or 1/tangent.
How can you remember these? Someone once told me about the phrase 'Co-Co No Go.' This helped me a lot! This helps me to remember that the cosecant goes with the sine and not the cosine. The tangent and cotangent are easy to remember since they share the same word 'tangent.'
So, let's go ahead and start with graphing the cosecant function.
Wow, this is interesting. We get mounds on the bottom and dips on the top. Well, this is to be expected. Since this is the reciprocal of the sine function and our denominator is now the sine function, we know that we will have an asymptote whenever the denominator, the sine function, equals 0. So, we have asymptotes at pi*n = 0, pi, 2pi, .... Also as expected, just like the sine function, we have the same standard period, or length of a cycle, of 2pi.
Moving on to the secant graph.
Hey, this looks a lot like our cosecant graph! And if we weren't careful we might mistake it for it. We have the same mounds and dips. The only difference is that these mounds and dips occur at different spots now. Just like our cosecant graph, our secant graph has asymptotes wherever our cosine function is 0, so we have asymptotes at (pi/2) + pi*n = pi/2, 3pi/2, 5pi/2, .... Also, because our cosine function has a standard period of 2pi, so does our secant graph.
Lastly, here is our cotangent graph:
Just like the other two graphs, our cotangent graph has asymptotes wherever our tangent function is equal to 0. Our tangent function equals 0 every pi*n spaces, so at 0, pi, 2pi, and so on. We see that our cotangent function does have asymptotes corresponding to these places. We also see that, just like our tangent function has a standard period of pi, so does our cotangent function.
We've covered all our graphs now. Let's talk about shifting our graphs, or transformations. We can easily make our graph move up, down, to the left and right by just adding or subtracting numbers from different areas of our function.
To move our graph up or down, we can simply add or subtract numbers from the end of our function. For example, the function csc (x) + 2 has a shift of 2 spaces up. If we wanted to move the graph down, we would subtract two from the end.
If we wanted to move our graph to the right, we would subtract numbers from our argument, our variable. So, the graph of cot (x - 2) is the graph of cot (x) shifted 2 spaces to the right. If we add 2 instead of subtracting, we would then have a shift of 2 spaces to the left
If we multiply our function by a number, it will change how wide the middle part is. For example, in the cosecant and secant function, this number will change how far apart the mounds are to the dips. Larger numbers make the mounds go lower and the dips higher thereby increasing the distance between the two. So, the graph of 4 csc (x) looks like this:
Look at the larger gap between the mounds and the dips. For the cotangent graph, it changes how long it takes for the curve in the middle to happen. So, the graph of 4 cot (x) has a less pronounced curve because it provides more space for the curve to happen:
One last transformation we can have is that of the period. If we multiply our argument, our variable, by a number, it will change our period from the function's standard period to the new period found by dividing the standard period by the number we multiplied. So, the period of csc (2x) has a period of 2pi/2 or pi. The standard period of the cosecant function is 2pi. Multiplying our argument by 2 changes this standard period to 2pi divided by the number we are multiplying. If the number we multiply is less than 1, then our period will increase.
Let's review what we've learned now:
Today, we looked at the graphs of the cosecant, secant, and cotangent functions. We define cosecant as the reciprocal of sine or 1/sine, secant as the reciprocal of cosine or 1/cosine, and cotangent as the reciprocal of the tangent function or 1/tangent.
We saw that the graphs of these functions shared the same standard period as the functions that they are reciprocals of. So, the cosecant has the same period as the sine function of 2pi, the secant has the same period as the cosine function of 2pi, and the cotangent has the same period as the tangent function of pi.
Also, these functions have asymptotes where the reciprocal functions have 0s. So, the cosecant function has asymptotes where the sine function equals 0, so every pi*n. The secant function has asymptotes every (pi/2) + pi*n. And the cotangent has asymptotes every pi*n spaces.
We can make transformations or shifts in our graph by adding or subtracting numbers in different areas. If we add or subtract numbers from the end of our function, we move our graph either up (add) or down (subtract). If we add or subtract numbers from the argument, the variable, we move our graph either to the right (subtract) or to the left (add). If we multiply our function by a number, then we increase the distance in the middle. If we multiply our argument by a number, then we change the standard period of the function by dividing by the number we multiplied. So, the function csc (2x) has a period of 2pi/2 or pi.
Following this video lesson, you will be able to:
- Define cosecant, secant, and cotangent
- Identify the graphs of these functions
- Recall where these three functions have asymptotes
- Explain how to make transformations to the graphs of cosecant, secant, and cotangent
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Graphing the Cosecant, Secant & Cotangent Functions Quiz
Instructions: Choose an answer and click 'Next'. You will receive your score and answers at the end.
The secant function has how long of a period?
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