Using Inverses to Evaluate

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SWBAT evaluate compostions involving inverse trigonometric functions.

Big Idea

How can you evaluate cosine when you are given arcsin x without finding the angle?

Bell Work

5 minutes

The students are asked to evaluate sin(arcsin 1/2). Students work in their groups to discuss this problem.  Some groups quickly say 1/2. I ask the group how they found the answer so quickly. The groups say that sine and arcsine are inverse so your answer is you original input. Other students will find the angle for arcsin 1/2 (which is pi/6) and then evaluate sin pi/6 to get 1/2.

For some students to understand I put sin(sin^(-1)(1/2))on the board. This helps the students see the structure of the inverse since the arcsine notation is new to the students. If students still don't understand we work the problem by find the value of arcsin 1/2 which is pi/6 and then find the sin(pi/6). Demonstrating the reason why the answer is 1/2. Even though this is similar to the problems we did with inverses in the Function and Piecewise Function Unit some students struggle to see how this is the same as algebraic functions. With more work students will see the connection.

Composition of Trigonometric functions

15 minutes

I want to formalize the process in the bellwork.  I put on the board

sin(arcsin x)=

Students are asked to find the answer and explain why this is true.

Is this the same for cos(arccos x) and tan(arctanx)?

I give the students some examples.  I ask the students to verify by working the problem out. I remind students that we have to work inside out. Some students notice that the second example does not give us back the angle in the original problem. Other students will argue that the angle must be the same because the 2 functions are inverses. I begin the discussion asking questions like:

  • Why do you think the angle is different?
  • What is the domain and range of y=arcsin x?
  • Is the original angle in the range of arcsine?

After discussing how this happens the students decide that the answer to arcsin(sin x) is x if x is in the restricted range. "Will this same rule be true for inverse cosine and inverse tangent?"

Many students will ask if they can just work the problem out like we did to verify the answer. Of course this is a method that works and can be used. As students become more proficient with trigonometric functions they will use the inverse properties to simplify.

I give the students a few more examples to work out in groups then share the results.

Evaluating abstract compositions

10 minutes

I now bring in an idea that is used in Calculus II (trigonometric substitution for integrals) and takes the work from numerical to algebraic.  The next problem is put on the board. Students are given some time to discuss the problem.  I move around the room looking at student work and making note of student reasoning.

As soon as I remove the numbers from the problem some students are confused. "How did you do the problem when there was a number instead of x? Can you do the same with this one?" Some students will be confused about what x>0 is telling them?  I question students with questions such as: What does it mean to say x>0? What does that tell you about the value of arctan (1/x)? Where in the coordinate plane will the triangle representing the arctan 1/x be located?

After about 4 minutes I ask a group to share their process on the board. Some questions I use to help with understanding include:

  • How do we know the angle is in quadrant I?
  • What does the x>0 tell you about the angle?
  • Once you identified the location of the angle could you find information about y and r?
  • Did any of you make a triangle to help? Put this on the board.

Once students have discussed the problem I give the class another example (page 2). After working the problem students share their answer.


5 minutes

As class ends I assign page 348, #55-69 odd from Larson "Precalculus with Limits"

Today students are given an exit slip. The problem requires the students to determine the quadrant and then find the sine in that quadrant.  I will use the slip to identify students that need more instruction with this topic.