The Unit Circle Day 2 of 2
Lesson 7 of 19
Objective: Students will be able to define trigonometric functions of acute angles.
Warm-up and Homework Review
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 - Unit Circle Day 2 which asks students to analyze an expression with a root in the denominator.
I also use this time to correct and record the previous day's Homework.
This investigation is the second of a two day lesson. By the conclusion, the students will have a unit circle with the most important ordered pairs. In the previous day's lesson, the students went through 45-45-90 and 30-60-90 triangles on a unit circle. Today the students cover both 90o and 180o. By now, students should be familiar with the fact that the sine and cosine represent the y and x values of an ordered pair on the unit circle. This should help them overcome the fact that there is no triangle formed at 90o. It is important to discuss the tangent of 90o. I have them discuss in pairs why it doesn’t exist. They have discussed dividing by zero a ton when it comes to domain and rational functions. This should come pretty quickly.
The next step is to fill in both the angle (degree and radian) and the coordinates of the unit circle. We do this as a class. I model 90o and 180o since that was just done writing the two versions of the angle on the line and the coordinates at the end of the segment. Next, I have the students identify the 45o angle and ask them to fill in the coordinates. The two remaining angles are the 30o and the 60o. Once we have gone over these angles, I have the students fill in the remaining circle reminding them that all of the remaining angles have the same reference angles as the ones on the angle. This final portion can be assigned as homework if time is short.
The goal of this section is ensure that the students know how to find less common trig ratios using a calculator. My students did this the previous year but it is important to make sure all of the students are on the same page before we get into modeling with trigonometry.
I introduce the first non-standard angle and have the students locate it on the unit circle. I ask for ideas how we should go about finding the cosine of this angle giving them a chance to talk about it in pairs (Math Practice 2). I don’t tell them in advance that they will need calculators. I manage the conversation, potentially asking leading questions; until the fact comes up that we don’t have enough information or skill. This is where I introduce the calculator.
The next goal is to introduce (remind) students how to change between radians and degrees on the calculator. Once they have had a short tutorial, the rest of these problems should be speedy.
Now I ask the students to find an angle given the sine, cosine or tangent. Again, I give the students time to talk about the first problem. My students saw this in Geometry last year. During the discussion, hopefully someone brings up the arc sine. They have done many inverse operations, in fact, the last unit focused on logarithms which is the inverse operation of an exponential. This is a good angle to take in introducing the arcsine. The other important thing to focus on is what the arcsine is actually asking. For example, in the first problem, it would be asking us “what angle has a sine of 0.7563”.
Now, we talk about the fact that these trig ratios are true for an infinite number of angles so we need to place a limit. I start with 0 to 2π. There will be two angles for each problem. This is a great review of the location of angles with the same trig ratio(s).
The first portion of this Homework asks the students to write the trig ratios for several of the most common angles. They have seen this before in prior lessons but repetition improves retention. The next several problems have the students use a calculator to find the trig ratio of an angle. It includes both degrees and radians. The final section asks students to use the inverse trig function to find an angle within specific parameters. The final problem asks the students to identify all of the angles that have a specific arctangent.
I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.
The Exit Ticket asks students to show that they can find an angle given the cosine ratio and a degree parameter.