Properties of Sine and Cosine

5 teachers like this lesson
Print Lesson


SWBAT define cosine, sine, and tangent ratios and to distinguish among them. Students will understand the meanings of the sine and cosine functions and their inverses.

Big Idea

Exploring the meaning of sine and cosine as ratios and as relationships. Trigonometry is just similar triangles—but better!

Lesson Opener and Goal Setting

10 minutes

The lesson opener asks students to recall the definition of the tangent ratio and to think of at least two more ratios that can be written between the sides of a right triangle.  (There are 6 all together, including reciprocals.)  The idea is to help students see that if they can write a ratio comparing the lengths of the legs of a right triangle, then they can write ratios involving the hypotenuse, as well.  In fact, they need to know these ratios to find the length of the hypotenuse or to find the length of a leg when only the hypotenuse is known.

This activity follows our Team Warmup routine, which is described in my Strategy folder.

 While students are working on the lesson opener, I complete administrative tasks.  These include taking attendance and noting which students have not completed their homework or brought required items to class. 


Following the lesson opener, I display the learning goals and agenda for the lesson using the overhead projector and review them briefly with the class.  I tell the class that today we will be learning about two more trigonometric ratios.  Together with the tangent ratio, the sine and cosine ratios allow us to solve for any missing side of a right triangle.   I remind them that they have already learned about the tangent ratio—which relates two sides of a right triangle—and the tangent function, which relates the tangent ratio to the measures of the angles of the triangle.   Similarly, they will learn about the sine and cosine functions today, as well as about their inverses: arc sine and arc cosine.

Introducing Sine and Cosine

25 minutes

Using the Properties of Sine and Cosine notes as a teaching aid, I lead the class in a discussion of the properties of the sine and cosine functions as well as of their inverses.   More information on my use of guided notes can be found in my Strategy folder.

Earlier in this unit, students used guided notes to summarize what they had already learned about the tangent ratio, tangent function, and arc tangent function in class.  Now, students complete the guided notes up front as I lead them in a discussion of the properties of sine and cosine, arc sine and arc cosine.  Since they are already familiar with tangent and arc tangent, the discussion reinforces and extends students’ understanding of a trigonometric function and its inverse.  The notes are designed to help students see how sine and cosine are related to tangent, and how arc sine and arc cosine are related to arc tangent. 

The notes are also designed to remind students that sine and cosine are ratios (which can easily be missed, since they are presented in the form of a decimal fraction in trigonometric tables and calculator displays).  For this reason, the discussion begins by writing ratios using the sides of a set of five right triangles.   Those ratios are then compared in a table.  As students write the ratios, I ask them to calculate the decimal equivalents and compare them to the values they obtain from the trig tables.  I don’t ask them to do this with every single ratio, just enough to make the point that the values obtained from the table are ratios, such as the students are writing themselves using the diagrams of the triangles.  This serves a second purpose by familiarizing students with the tables. 

While some students will know how to find the trigonometric ratios using their calculator (and I do not try to stop them), I encourage students to use the tables at this point, because using the tables helps students to see that the trigonometric functions are relationships between an angle and a ratio of sides (MP2).  Too often, students think that the point of math is to get an answer (which calculators are good at), rather than to understand relationships.  By putting off the time when students rely on calculators to obtain trig ratios, I hope to give them the opportunity to form their first understandings of trigonometry in terms of relationships. 

Next, the relationship between the acute angle of a right triangle and the associated sine and cosine ratios is examined by plotting ratios against angles in a graph (MP2).   As I lead the students in completing the guided notes, I ask questions and encourage students to do the same, so that the activity is conducted like a whole-class discussion.   Possible questions:

  • What is the difference between the sine (or cosine) ratio and the sine (cosine) function?  How are they related?
  • What is the difference between the sine (or cosine) and arc sine (arc cosine) functions?  How are they related?
  • What is the relationship between the sine and cosine functions?  How can you explain this? (MP2)


Finally, the definitions of the arc sine and arc cosine functions are presented, together with a pair of examples.  The point of the examples is to walk students through the process of finding a ratio (which must obey a particular definition), then finding the associated angle. 

The final example in the notes shows the students how to use trigonometric ratios to find the missing sides of a triangle.  I do not give the students this example at this time, because I want them to reason for themselves how to apply the properties of sine and cosine (MP1).  The example can be completed later to summarize what students have learned by themselves or to show a recommended method when necessary to clear up confusion.  It can also be omitted.

Applying Properties of Sine and Cosine

10 minutes

Before class, I print the resource for this activity.  I make one copy for every two students and cut into half-sheets.  You may want to make a few extra copies, in order to allow students to start over with a fresh sheet if they go down the wrong path.  I distribute the half-sheets to the teams.   I tell the students that they are work in pairs to complete the problems. 

I use a Kagan Structure (Rally Coach).  My students learn the rules and roles for this mini-activity at the beginning of the school year, so it is a classroom routine.  More information on my use of this structure can be found in my Strategy folder.  The instructions are in the presentation.

As students are working, I circulate around the classroom.  

Lesson Close and Homework

8 minutes

I display the lesson close question on the front board using the overhead (the lesson close is in the presentation).  I have the students brainstorm in pairs, then in teams, before writing their answers in their learning journals.  The purpose of the learning journal is to encourage students to reflect and to provide individual accountability.  Time permitting, I also ask one student from each team to write a team answer on the white board.  This gives me feedback on what students learned.


For homework, students complete problems 3-4 of the handout that was assigned in the last lesson..