Algebra II : Unit #9 - Trigonometric Functions : Lesson #2

Trigonometric Ratios

Objective: Students will be able to define trigonometric ratios of acute angles.
Standards: HSF-TF.A.2 MP1 MP2 MP3 MP7
Subject(s): Math
60 minutes
1 Warm Up and Homework Review - 10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. This lesson’s Warm Up- Trigonometric Ratios Warm Up asks students to solve a traditional Pythagorean Theorem word problem using a ladder against a wall.  The Pythagorean Theorem will be important to today’s lesson so this is a great mini review.

I also use this time to correct and record the previous day's Homework.  

Remembering An Old Skill
Connection to Prior Knowledge

One simple way to challenge students (Math Practice 1) is to present them with a skill they learned in previous years with no prior warning.  When presented with this context problem with no warning, about 40% of my students caught on right away.  The rest required a certain amount of coaching either from their peers or myself.  At this point in their mathematical journey, they need to be able to fluently recall and use prior skills.  This is going to be fundamental as they head to either the PARCC or the SBAC test in their junior year.  Many of the skills they will use on this test will be older skills.

The issue I noticed on the solving of this problem centered on thoughtfulness.  The only students who didn’t get it right, misplaced the hypotenuse like in this Warm Up.  We went over this problem in class and students quickly rectified the problem.

2 Trigonometric Ratios - 25 minutes

I introduce this lesson by starting with a right angle, making sure the students remember the terminology legs and hypotenuse.  I then place the right triangle in a circle centered on a coordinate plane.  The intersection of the circle and the triangle is called (x,y).  I then ask the students to talk in pairs and determine the length of the legs of the triangle.  They should figure out that the horizontal side is the same length as the x value of the intersection and the vertical side is the same as the y value.  We will call the hypotenuse r for radius.  This will make a difference when we talk about radians. 

My students saw the trig ratios in Geometry the previous year.  Here we review their basic geometric definition using the position of angle ө.  We also add the algebraic definition using x, y and r.  Again, this will be important as they head into radians.   Please watch my Video Narrative on prior knowledge.

Using a coordinate to represent (x,y)on a circle, we then find the sine, cosine and tangent ratios.  They need to get r to find sine or cosine but I don’t tell them in advance (Math Practice 1).  I walk around and give hints or individual aid to students that are really struggling.  Pythagorean Theorem was the warm up so it shouldn’t be entire unfamiliar to them. 

Next, they are asked to find sine, cosine or tangent given one of the ratios.   This is similar to the previous problem except that they have to decode a trig ratio onto the graph, find the third side and then find a new trig ratio (Math Practice 2).  This is a more complex task than the previous one.  Again, I give the students an opportunity to work on this problem with no prompting.  Scaffolding can be provided individually as necessary or to the group if a large number of them are floundering.  Another example is provided to solidify the concept.  

3 Special Right Triangles - 13 minutes

The remainder of the lesson focuses on the trig ratios found on the two special triangles 30-60-90 and 45-45-90 (Math Practice 7).  One of the Pre-Calculus teachers at my school told me that one of the best things that students could bring to his class would be a larger understanding of trig, particularly as it relates to 45-45-90 and 30-60-90 triangles.  I am choosing not to have my students memorize the basic trig functions but we will continue to visit these special triangles and by the end the students should have a solid understanding to those triangles.  It would be good to talk about why these two are particularly important.  The more connections students can make, the better they will understand them.  

Please see the PowerPoint for detailed presentation notes.

4 Homework - 0 minutes

The Homework is a practice of the skills they learned in class.  The best way for students to memorize the trigonometric ratios is to use them.  This assignment will help them do that.  It also asks the student to reproduce and explain how to find the trigonometric ratios off of the two special triangles.  

5 Exit Ticket - 3 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

Today's Exit Ticket asks the students to find sine, cosine and tangent given the (x,y) coordinate form on a graph.