Solving Triangles with Trigonometry

The lesson opener asks students to find the missing side of a 30-60-90 right triangle in two ways: by using the properties of special triangles and by using trigonometric ratios.  This permits a comparison of methods and reminds students where trigonometric ratios come from (MP5).

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 practice solving triangles using all three trigonometric functions that they have learned.   

Before class, I print the resource for this activity.  I make one copy for every two students and cut into half-sheets.  I usually 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 to work in pairs to complete the problems. 

I use a Kagan Structure (Rally Coach).  More information on my use of this class routine can be found in my Strategy folder.   The instructions are in the presentation.

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

• Students may have difficulty identifying the ‘opposite’, ‘adjacent’, and ‘hypotenuse’ sides of a right triangle.  I advise students to start with the hypotenuse: it is opposite the right angle and is never called ‘adjacent’ (although it is always adjacent to the acute angle).  To identify the legs opposite and adjacent to the given angle, I imagine standing in the angle of the triangle with one arm over each side, like a boxer draping his arms over the ropes of a triangular ring.  The side under my Armpit is Adjacent (the side under my other armpit is the hypotenuse, which is never called ‘adjacent’).  The side across from the angle is Opposite.
• Students may look for patterns to help them decide when to multiply and when to divide the given side length by the trigonometric ratio.  For example, a student may notice that you multiply by the tangent ratio when the unknown side is opposite the acute angle but divide when it is adjacent.  The problem with this method is that it is not generalizable, so students must recognize and memorize a different rule for every trigonometric ratio.  I point this out to students and ask them to look for a different method.
• Students may make algebra errors.  For example, when the given length, c, the side to find, x,  and the corresponding trigonometric ratio, k, are in the relationship c/x = k students may multiply both sides by c or by x, rather than take the reciprocal of both sides.  I encourage students to write trigonometric equations in the form c/x = k/1 or x/c = k/1 to help them to see that they are solving a proportion.  

Students should have time to begin on their homework in this lesson, which gives me a chance to answer individual questions.  With 5 minutes remaining in class, 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 from the lesson.

The homework assignment for this lesson is provided in the unit syllabus.