Formalizing Properties Known Informally

Learning Target for identities

This is the beginning of a unit where we formalize trigonometric identities that we have been using such as the reciprocal identities. The students will also analyze mathematical arguments (proofs) and develop mathematical arguments.

I quickly review the trigonometric ratios from the unit circle and the right triangle. I make a chart and put each trigonometric function in the chart. I have 2 columns that ask these questions?

• If you have a right triangle, how do you find the value of the trigonometric function?
• If you have an ordered pair (x,y), how do you find the value of the trigonometric function?

I pick a student to write the class answers on the board.  This student chooses other classmates to answer the questions.  The only requirement is the "teacher" must pick a different student for each space.

After students have the "rules", students find values of the trigonometric functions when they know the value of one function. Students usually determine x=4 but forget -4. As they put up answers I say "How did you know x=4 not -4?" This leads to students determining there are 2 answers for most of the functions.

Before we get too far into this lesson, I want to make sure the students develop a common understanding of the term identities. As a start, I allow students to get out their phones to look up a definition.  I explain that I want the mathematical definition.  Most of my students use the site Math is Fun when I let them search in class.

Once students have searched, I ask a student to read the definition. I write the definition on the board so that we can discuss the definition.  Some ideas that I want students to understand is that an identity is an equation and that it always true. When we discuss how an identity is an equation I talk about how there are 2 parts equivalent parts of an equation and that one side can replace the other side in another equation or expression.

After clarifying the definition I ask students to think of identities they know:

• What is an identity we know right now? Some students will say the Pythagorean Theorem. When this comes up we go back to the definition and say is this true for all values of a, b and c or just certain values? Student may take one of the properties from the bell work like sin(theata)=y/r. We look to see if this is true for all values of y and r.  Of course students need to understand that y is a coordinate and r is the distance from the origin to the point (radius of a circle) for this property to be true. 

I put a quote on the board. I ask the following questions about this quote:

• What is this statement saying we will do with the trigonometric functions? This questions is to help them understand the quote.  The students need to understand that we are going to write equations with trigonometric functions so we can replace an expression with an equivalent expression.
• Do you already know a trigonometric identity that we could write? Students have been using the reciprocal properties as they worked with the unit circle and when we graphed secant and cosecant. If students do not think about the reciprocal property I will prompt them by saying what do you know about secant? How can we write mathematically? (See page 2 of resource)
  

I give each student a piece of white paper.  This sheet is for students to develop a reference sheet for the identities we will develop in this unit (see sample reference sheet). The students are told to title the page. I explain that this will be a sheet they use to remind them of important identities we will learn during the unit. Some students will not want to write the properties out.  I try to explain that the sheet will help them remember properties once we get more.  Even if they know these identities by having them on a sheet it will help jog there memory when they are working. I also allow this to be used sheet to be used on quizzes.

Students like a rational for new material. For this lesson I explain that I want to be able to determine the value of all the trigonometric functions without needing to use a triangle or using the parameters x, y, and r.

I put up the expression sin(theta)/cos(theta) on the board.  Can we simplify this to a single trigonometric function?  I let the students work on this for a couple of mintues.

When I see confusion I will clarify by saying:

• How could I use the rules from the bell work to help simplify this expression? 
• What does sin(theta) equal and cos(theta) equal? (page 2)
  

Students will start to see what to do.  Students work to simplify the expression.  After 2 minutes I ask a student to share their process. (page 3)

Once the students see sin(theta)/cos(theta)=y/x students realize that y/x is tan(theta). 

Can we now get a formula for cotangent in terms of sine and cosine?

Students are given another example to work. I tell them to use the identities on the reference sheet to find the other trigonometric functions.

Given  sin(theta)=sqrt(5)/3 and cos(theta)=-2/3 find the value of the other trigonometric functions.

Students work on this and share their results. Students discuss how they found the values of each function. The question I usually start a discussion with is "what function value did you find first?"

The function that is the most difficult for the students is finding the value of tangent. The struggle is in simplifying the fractions.