Lesson 6

The Pythagorean Identity (Part 2)

Lesson Narrative

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 warm-up 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 in which 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).

Learning Goals

Teacher Facing

  • 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.

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.

CCSS Standards

Addressing

Building Towards

Glossary Entries

  • Pythagorean identity

    The identity \(\sin^2(x) + \cos^2(x) = 1\) relating the sine and cosine of a number. It is called the Pythagorean identity because it follows from the Pythagorean theorem.

  • period

    The length of an interval at which a periodic function repeats. A function \(f\) has a period, \(p\), if \(f(x+p) = f(x)\) for all inputs \(x\).

  • periodic function

    A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).

  • unit circle

    The circle in the coordinate plane with radius 1 and center the origin.