Graphing the nonsinusoidal trigonometric functions Day 2 of 2

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SWBAT graph by hand secant and cosecant functions.

Big Idea

How can the graphs of sine and cosine be used to help find the graphs of secant and cosecant?

Bell work

5 minutes

When I graph secant and cosecant I use the relationship these functions have with sine and cosine.This idea gives me a chance to remind students about reciprocals. In other words if you have a number that is between 0 and 1 the reciprocal of that number is greater than 1. Some students are still working on understanding this relationship. Connecting old ideas with new ones will help students see the importance of ideas they have learned before.

I begin the class by having students graph one period of y=sinx. The students will quickly make a graph. I have a student put the graph on the board.

Graphing Cosecant

15 minutes
Now that students have a graph of y=sin x. I am ready to use the graph of sine to help graph cosecant. I use the following questions to develop the graph of y=csc x:
  • How are the functions sine and cosecant related? Students remember they are reciprocals.
  • How will this relationship show up on a graph?  I may put as the students to find the value of sin pi/6 and csc pi/6 to help students see the relationship.
  • When sine is 0 what is the value of cosecant? How do we show this on the graph?
  • When x is between 0 and pi sine is a positive number between 0 and 1. What will cosecant values be between 0 and pi? I want students to understand that the values will be 1 or greater.
  • What will the values of cosecant be when x is between pi and 2pi?
Some student may say the graph flips over. I ask students to explain this reasoning. The explanation includes discussing how the the graph of sine divides at the x axis and the the point above the x axis go above 1 and the point at pi/2 does not move. The points below the x axis go below -1 while 3pi/2 does not move.
When using SMART Notebook, I can graph y=sin x and y=csc x trace parts of the sine graph then rotate the trace to show how the point are similar. I ask why the graph does not stop at  pi? We will put some points close to pi in the calculator to show how the value of cosecant continues to grow as x gets closer to pi. 
Once we have analyzed the graph I ask: How can you use the graph of sine to help you graph cosecant? Are there some key points on sine that will help you graph cosecant? My goal is for students to realize they can find the maximum, minimum and midline values of sine and use those to identify the asymptotes and where the graph will be at a turning point for the pieces of the graph.

Students are given 3-4 minutes to graph y=csc x +2. I move around and ask questions such as "How can sine help you graph? When is cosecant undefined? When sine is at a maximum what is happening to cosecant? How does the 2 change the graph?"  I have student go back to the development to help them do this graph. I then have a student put a solution on the board for discussion. Some question that I ask include:
  • Did you use sine to help you graph?
  • How did you locate the asymptotes?
  • How did you know where the graph was positive?negative?

Graphing Secant

15 minutes

I am ready to see if students will connect cosine and secant as I did sine and cosecant I put up the equation y=sec x on the board. I want students to think about what we just did so I ask: How could we graph this function? Consider this question on your own for a minute. After about a minute I have students share with a neighbor their ideas. I listen to the conversations to see if students connect secant and cosine. After a couple of minutes I ask someone to share out the ideas.

When the class begins to say graph cosine and do like we did not the cosecant. I will ask a student to show me what they mean. I sometime have 2 students work together to help with the graph.

As the students are working I say,
  • Why are we using cosine to graph secant?
  • So when cosine is 0 what is the value of secant?
  • Will the graph look like secant?
  • How are the graphs of secant and cosecant different?
Once the graph is on the board, I let the student analyze and verify the accuracy of the graph. We correct any mistakes. I put the graph of secant and cosine on the board so students can see that the sketch is the same the same. Some students need to have verification of what students say so using Desmos is a great way to quickly verify the answer.
As with cosecant, I have students try a transformed secant graph (y=3sec x). Students work om the graph and then share their results with each other. One person will put the graph on the board.


5 minutes

I give student a few problems to graph on as independent practice. Today pick page 337, #16, 22, 27,and 35 from Larson's "Precalculus with Limits, 2nd ed."

Today student are asked to answer this question. Tangent, cotangent secant and cosecant do not have an amplitude why not? Many students think that the coefficient in front of the trigonometric function is the amplitude but the amplitude is the maximum distance from the midline that a function value will be. Only sine and cosine have maximum and minimum values so they are the only trigonometric functions with an amplitude.