Today's Bell Work reviews the definition of an inverse function. We have worked with inverses in the Function and Piecewise Function Unit and in the Exponential and Logarithmic Function Unit. As we move through the next 2 lessons students will see we must limit the domian of sine, cosine and tangent so that we can have inverses.
The students share how they found the inverse. I then ask the students to remember how y=x^2 inverse is not a function if we use all real numbers as the domain. I share the graph of y=x^2 we then discuss how to limit the domain.
This brief review helps set up how we will proceed through this lesson
Each student receives Inverse Trigonometric Functions worksheet. As a class we divide the work up so that 3 different groups find the values for the tables of sine, cosine and tangent. Students share the results so students can find the inverse of the functions and then answer the questions for each function.
Students work for about 10 minutes after the tables are presented. I bring the class together to discuss the activity. In the table I do not use y=arccos x or y=cos^(-1) x since the table does not represent a function.
I start with question 2 on page 2. I ask students to answer the question and explain their reasoning. We move to question 3. How can we make the inverse a function. Some students will say only use half the circle. "Okay, what half?" The students reason it makes sense for the angle to be between 0 to pi to have all values of cosine to have an angle connected.
Many students will have confusion with the idea that the range for the inverse is really the angles that give the specified value for cosine. Students have difficulty with this idea of inverse because they have not worked with trigonometric functions as long as they have worked with algebraic functions. Algebraic functions do not use numbers such as pi/3 and students have still not seen how this can be an actual number. They it will take more time working with these types of numbers to become proficient with them. I will show student that when we say pi/3 that is the exact value which can be approximated by 3.14/3 so it is a little over 1.
Understanding the notation for inverses is important. Student quickly give cos^(-1) since we used this notation when we found angle values in the Solving Problems Involving Triangles Unit.
Another common notation is arccos x. I explain that this notation comes the idea of the unit circle if the radius is one then the arc length is the same as the angle in radian. One place to find more information about the term is:
There is also a third notation, acos x, that is used in some advanced college math class and in some engineering classes. I feel it is important to let students see the different notations. Being introduced to the notations will help when they see the notation later.
I now put y=arccos sqrt(2)/2 on the board for the students to evaluate. Some students want to find 2 answers, since that was what we did in the previous lesson. I discuss the difference between this equation and cos(theta)=sqrt(2)/2. We discuss how arccos x is a function with a defined domain and range where the previous equation is wanting to know all angles that have specified value.
As part of our literacy initiative I am working on helping my students develop an "inner voice" as they read mathematics. I demonstrate my "inner voice" as we discuss problems. For this lesson I explain the question I ask when I see the inverse notation:
When I see a problem like the one above, I ask myself what angle gives me a value of sqrt(2)/2?
I ask the class how the tools we have developed in this unit will help us find the answer. Some use the unit circle to find the answer while others will draw triangles to determine the angle.
I always work with the inverse of cosine first because students see how to limit the domain quickly. Sine and tangent are a little more confusing. I begin by looking at the work from the activity. Students see that we need to limit the domain to make the inverse a function.
I ask students to think about how we need to limit the domain. Some will say use quadrant I and III others will say quadrant 1 and IV. I ask students to defend the choice.
I want students to consider how we would like to keep the domain continuous.I have the students determine the sign of sine in the coordinate plane. I make the comment that mathematicians like to make the restrictions so we don't have breaks in the function and want to have 0 as a possible angle. The students see that Quadrants I and IV must be where we limit the domain of sine. I continue to say mathematicians want to write the domain as simply as possible. So instead of saying for 0 to pi/2 and 3pi/2 to 2pi we write the quadrant IV portion as negative angles. Through this process students determine the domain of sine is -pi/2 and pi/2. Of course this means that the range of arcsine is -pi/2 to pi/2.
Once we limit the domain for sine the students see how tangent has the same limits.
I then ask, "In what quadrant will we not have inverses?" I want students to realize that the answers for inverses will never be in quadrant III. This idea helps some students with inverses. (I have also used the idea that when finding the inverse of a trigonometric functions angles in quadrant III are in "no man's land" this is another way for students to remember where the angles are located)
As this lesson comes to a close, I will choose an appropriate problem for my students to complete as a Exit Slip. I will ask students to put the answer on their worksheet so I can review the answer as we end. I keep track of students that do not get the correct answer to work with during the next class.