Graphing the nonsinusoidal trigonometric functions Day 2 of 2
Lesson 8 of 13
Objective: SWBAT graph by hand secant and cosecant functions.
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.
- 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?
- Did you use sine to help you graph?
- How did you locate the asymptotes?
- How did you know where the graph was positive?negative?
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.
- 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?
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.