Applying the Tangent Ratio

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Objective

SWBAT use the tangent ratio to solve for the missing sides of a right triangle. Students will understand the relationship between the measure of the acute angle of a right triangle and the lengths of the sides opposite and adjacent to the angle.

Big Idea

Students extend the concepts they have already used to solve special right triangles to right triangles of any angle—Behold! Trigonometry.

Lesson Opener and Goal Setting

15 minutes

Present the lesson opener.  This activity follows the Team Warm-Up routine, which is described in my Strategies folder.  The opener for this lesson (found in the slide show) asks students to find missing sides of two right triangles (MP1).  The first triangle is a special right triangle with which students are familiar.  The second has a 40 degree acute angle.  In the previous lesson,
students used a virtual tool to build a table of tangent ratios, including the ratio for a triangle with a 40 degree acute angle (MP5). 

My goal is for students to understand that they can use the same methods for solving the 40 degree right triangle as they use for the familiar 45-45-90 triangle.  As I circulate through the
classroom, I offer hints and encouragement.  If necessary, I call for the class’s attention and lead a quick class discussion on how students can solve the second triangle.  Suggested questions:

What approach did you use to find the missing side of the first triangle?  Could that approach work for the second triangle?

What makes the second triangle harder to solve than the first? 

If there were one piece of information they you could know about that second triangle that would help you to solve it, what would it be?

What was the point of yesterday’s lesson? Could you use what you learned yesterday to solve the second triangle?


 
I display the agenda and learning goals for the lesson while distributing the Trig Tables (Trig Tables.docx).  (I created the Trig Tables using Microsoft Excel.)

I tell students that mathematicians have studied the relationship they investigated yesterday for 2000 years, because understanding it means that you can find the ratio between the lengths of the legs of any right triangle if you know one of its acute angles.  Knowing that ratio of side lengths, you can find the length of one leg when you know the other.  I tell students that this way of thinking is at the heart of trigonometry, which is one topic in mathematics that most people have no trouble seeing as useful.  I tell students that they already know the basics of trigonometry: they have been using that way of thinking whenever they solve special right triangles.  I tell the students that the ratio between the legs of a right triangle is called the
tangent ratio, and it is one of three ratios that are published in a typical trig table.

Practice Applying the Tangent Ratio

30 minutes

Orient student to the trig tables (5 minutes).   Ask students to get calculators from the Resource Center if they do not already have one, as they will need one throughout the lesson.  Ask students what the tangent ratio should be for a 45-45-90 triangle, then demonstrate how to use the table to find the tangent of 45 degrees.  Repeat the question for a 30-60-90 triangle, and point out that the answer depends on which leg you know and which you are trying to find.   Have students find the decimal fractions that correspond to the ratios they know (0.577 or 1.732) and confirm that they can find that ratio in the table. 

Tell students that it is good practice to write trig ratios to the thousandths place in order to have the required precision (MP6).

 

Have students work in pairs and teams to practice applying the tangent ratio (20-25 minutes).   Before class, print the resource for the activity.  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.  Distribute the half-sheets to the teams.   Tell the students that they are to work in pairs to complete the problems (MP1). 

 

I used 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 can be found in my Strategy Folder.   The instructions are in the slideshow.


As students are working, circulate around the classroom.  Common problems to look for:

  • Students may indiscriminately multiply the length of the leg that is known to find the unknown side length.  Remind students that the tangent ratio, like all ratios, has a certain direction (length of the leg opposite the given acute angle to the leg adjacent to the given angle).   This direction has to be respected when using the tangent ratio. 
  • Students may identify patterns to help them decide when to multiply or divide by the tangent ratio that are not based on a true understanding of the meaning of the ratio.  For example, they may notice that when the unknown side is opposite the given acute angle, you multiply by the tangent ratio, and when the unknown side is adjacent, you divide.  Memorizing rules like this will lead to trouble down the road, since they will have to memorize more rules when they learn to use the sine and cosine ratios.  For my thoughts on which student-developed methods are valid and which are flawed, see the video that accompanies the next lesson, Applying Tangent and Arctangent. 

At the end of this activity, I collaborate with students in awarding ‘team points’.  This is a class routine.   

Lesson Close and Homework

8 minutes

Have students reflect on the lesson in pairs (3 minutes).  Display the closing question and ask students to brainstorm answers in pairs, then as a team, before writing an answer.  Students write their answers in their learning journals.

Homework for this lesson is assigned in the syllabus for the unit.