This is the second of two lessons focusing on the Pythagorean Identity. The goal of this lesson is for students to deepen their understanding of the connections between \(\cos(\theta)\), \(\sin(\theta)\), and \(\tan(\theta)\) for an angle \(\theta\) on the unit circle (MP7).
The warmup invites students to identify the sign of the three trigonometric values in each quadrant, which allows a reintroduction of tangent and a possible way to interpret the value of tangent as a slope. Students then consider how to use the value of cosine in quadrant 4 to calculate the values of sine and tangent at the same angle. This work prepares students for a card matching activity where they determine if particular values for cosine, sine, and tangent are possible or impossible in different quadrants. For possible matches, students practice using the Pythagorean Identity to calculate the two unknown trigonometric values. Throughout the matching process, students trade roles explaining their reasoning and critiquing the reasoning of their partner (MP3).
Lesson overview
 6.1 Warmup: Math Talk: Which Quadrant? (10 minutes)
 6.2 Activity: Andre's Calculations (10 minutes)

6.3 Activity: Card Sort: Where's the Point? (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 6.4 Cooldown: \(\sin(\theta)\) and a Quadrant (5 minutes)
Learning goals:
 Critique (in writing) a strategy using the Pythagorean Identity for identifying the value of sine and tangent from the known value of cosine and the quadrant of the angle.
 Use the Pythagorean Identity to determine the value of all 3 trigonometric ratios given the value of 1 to start from and the quadrant of the angle.
Learning goals (student facing):
 Let’s use the Pythagorean Identity.
Learning targets (student facing):
 I can use the Pythagorean Identity to find the values of cosine, sine, and tangent of an angle if I know one of them and the quadrant of the angle.
Required materials:
 Preprinted slips, cut from copies of the blackline master
 Scientific calculators
Standards:
 This lesson builds towards the standard:CCSS.HSFTF.C.8MS.FTF.8
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.