Trigonometric Ratios of General Angles

3 teachers like this lesson
Print Lesson


Students will be able to find the trigonometric ratios of any angle on a coordinate plane.

Big Idea

It doesn't matter how many times you turn around or how dizzy you get, you can still find a trig ratio.

Warm-up and Homework Review

5 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. The Video Narrative explains this lesson’s Warm Up- Trigonometric Ratios of General Angles which asks students to compare and contrast the sine and cosine for two isosceles right triangles.

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

Introduction to Trig Ratios in all Quadrants

7 minutes

The first task asks the students to find the trig ratios of an angle that lies in the third quadrant.  This is the introduction to trig ratios of angles that are bigger than is possible in a right triangle.   This is one of the strengths of angles on a coordinate plane. I give the students some time to talk in pairs and then we discuss this as a class.  The goal is for the students to identify that we can always make a right angle given the x-axis and leads us directly to the concept of reference angles. 

Finding Reference Angles

10 minutes

The next goal is to define and practice finding reference angles.  We review the trig ratios introduced the previous day.   It is important for the students not to feel like they need to memorize these random variable ratios but to relate opposite with the vertical side of the triangle which is the y, and adjacent with the horizontal side which is x.  Some modeling helps a bunch.  

Finding Trig Ratios in All Quadrants

10 minutes

The next task has students attempt to find the trig ratios of an angle in the third quadrant.   Again, I give them the opportunity to work on this by themselves or in pairs.  Most of them will immediately give all three ratios as positive numbers.  At this point, I ask them how we should deal with the fact that both the x and the y value are negative.  How much guidance needs to happen here depends on the class.  The goal is to get to the point that students see that the sign of the x and the y affects the ratios.  Now, I have the students make a general chart of the sign of all trig ratios around the coordinate plane (Math Practice 7).  The students work with their partner to decide on the sign of each trig ratio in each quadrant.  If they are unsure, I model finding the first quadrant’s signs on the board.  It is important to be careful not to do this at very beginning of the activity as then the students won’t have the opportunity to figure out how to do it themselves. 

 After an adequate amount of time, we finalize the chart on the board.  It is important to note that this chart is NOT meant to be reference sheet that the students use to find the sign of each ratio as they go.  It is more of a tool to build understanding.  Next the students are asked to practice this skill by finding the ratios of several trig problems.  I may add a problem or two as needed for your students.  

Finding the Quadrant of a Trig Ratio

10 minutes

The concluding activity asks the students to do a small investigation extending a type of problem from the previous assignment to include angles of measure greater than 90o.  There will be two answers to the question “What quadrant would this angle be in?”  Some students may pick up on this while others may only find one.  This leads us to needing further information which in this case is that our angle exists between 0o and 180o.   From here I allow the students time to figure out the solution.  There is then an additional practice problem to solidify the skill.


The goal of this Homework is for students to practice finding the proper trig ratios for any angle.  This skill is crucial to student success in the remainder of this unit.

 This assignment was created using Kuta Software, an amazing resource for secondary mathematics teachers.

Exit Ticket

5 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 determine the whether the cosine of an angle is positive or negative.