Co-Function Identities

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SWBAT use the co-function identities to simplify expressions and prove identities.

Big Idea

Through reasoning students can verify identities for trigonometric functions.

Bell Work

10 minutes

As the students arrive, a verified identity is on the board for the students to review.  I ask students to spend a couple minutes discussing the identity in their groups. I expect the students to determine how each line of the proof was generated.

I bring the class together to discuss the process.

  • What was done in the first step?
  • Why did the person decide to work on the left? (I may comment here that it is usually easier to make move from an addition expression to a multiplication expression when dealing with trigonometric functions)
  • What was the reasoning behind the next step?

I continue questioning the class about the steps. As the students explain their observations I write their reasons beside the step on the board (see page 2 of bell cofunction).

After analyzing the process I ask "Was the process easy to follow? Why or Why not?" This question gives me an idea of how students organize their thoughts. 



Exploring Co-Functions

20 minutes

I let students work on the Complementary Angles worksheet for about 10 minutes. This worksheet has students find the trigonometric values of complementary angles and then determine which function values are equal. I remind students to have their calculators in degree mode for this activity. I used degrees in the activity because students will have an easier time noticing that the angles in the tables are complementary.

As students work I move around and answer questions. The main question I expect is, "How do I input cotangent, secant and cosecant into my calculator?"  We discuss how cotangent is the reciprocal of tangent. I demonstrate that cot 15=1/tan 15.

After students have worked for 10 minutes, we go over the Questions 2-3. On Question 3, I ask the students to focus on the first table.  We will work together to write down all the function values that are equivalent.  Usually, my students see the pattern. I ask, "If I know the value of sin 75 what else will I know?" Students quickly to say cos 15.

I then discuss why the functions cosine, cotangent and cosecant have these names.  I say something to the effect that the sin theta has the same value as the cosine of the complement of theta. I write this statement on the board so students can think about the statement.  We then write a similar statement for tangent and secant.

I know that some students are still a little confused and I need to develop a method for writing the complement of an angle.  I draw a right triangle on the board and have the students determine sin A and cos B. This helps students see that the ratios are equivalent.




Writing the Co-Function Identities

5 minutes

At this point in the lesson, I want to formalize what has been discovered during the Complementary Angles activity. I ask, "What do you know about angle A and angle B? Why?" I expect students to say they are complementary. I expect them to use one of two different ideas to explain their choice. Some will say all the angles add to 180 degrees and angle C is 90 so angle A and angle B add to 90. Others remember that the acute angles of a right triangle are complementary.

I then ask "how can you compute the complement of an angle?" Once the students say 90-the angle I will write <A=90-<B and <B=90-<A. Since we know that sin A=cos B, I can replace <B with (90-<A) correct.  We also write the expression in terms of angle B to make sure students see both equations.

In work related applications, it is often necessary to work with real number values instead of degrees as angle measurements. This is a good point to remind students that we have 2 ways to measure angles, degrees and radians. I will say, "We have just written the complementary angles in terms of degrees. How can we write the expressions in terms of radians?" After thinking for a minute students remember that 90 degrees= pi/2. I expect that my students will derive the correct expression.

After this prep work it is time to write the co-function identities on the students' reference sheets. I will put up the right side of the identity and a student will tell me what the expression is equivalent to so we can finish the equation. I randomly pick students to give the class the correct equivalent expression.

Using the Co-Function Identities

15 minutes

Just because the students have developed an identity does not mean they will be will be able to use it to solve problems. So, I will now give students a problem that requires them to find the value of trigonometric functions. My students are given time to work the problem on their own. After a few minutes, I will ask students  to display their answers for others to review. Then, we will talk about the processes that students used in this problem. I expect some of my students to have difficulty finding sine and cosine. At this point students have multiple methods to determine sine and cosine. So, having individual students share how they determined the value of sine and cosine will help all of the students to make connections.

Most students will use the Pythagorean Identities to find secant or cosecant and then determine sine and cosine. The issue students forget is usually determining whether secant and cosecant are positive or negative. I always ask how we know if a function value is positive or negative. I will also draw diagrams and put the terminal side of the angle in the appropriate position on the coordinate plane.

After working with numbers we move verifying identities involving co-functions. I give the students an identity let the students discuss how to verify with their groups. We then put a solution on the board. The co-function properties are not used in identities as much as the previous identities. These identities will be used when we find the sum and difference identities.


5 minutes

Students are assigned page 385 #20, 25, 33, 34, 35 from Larson's "Precalculus with Limits"

Problems 20, 25 and 35 review other fundamental identities and help students develop using algebra skills such as subtracting fraction. Problems 33 and 34 have students us the co-function properties.  

I like to give students problems that include old topics along with the new material. Many students are used to getting homework on just new material so they only look at the new identities to solve the assigned problems.  I want students to consider all the identities every time they start a problem. I remind them that every new set of identities increases our ability to solve problems.