Lesson 5

The Pythagorean Identity (Part 1)

Lesson Narrative

The goal of this lesson is for students to bring together what they have learned so far in this unit about points on a unit circle, right triangles, and trigonometry to establish the Pythagorean Identity. Students will continue this work in the following lesson.

In a previous lesson, students concluded that for any point \(B\) in the first quadrant whose distance from the origin is 1 unit, the coordinates \((x,y)\) of the point can be written as the cosine and sine of the radian angle corresponding to point \(B\).

Unit circle centered at the origin with right triangle drawn and cosine and sine labeled 

In this lesson, students first recall the connection between the coordinates of \(B\) and values of cosine and sine. Next, students expand this understanding to the other quadrants, viewing cosine and sine as coordinates of a point on the unit circle rather than as lengths of right triangle sides. This is an important transition step as students progress toward thinking about cosine and sine, and later, tangent, as functions.

Building from the equation of a circle and the coordinates for any point on the unit circle, such as \((\cos(\theta),\sin(\theta))\) for point \(D\), leads to establishing the Pythagorean Identity, written as \(\cos^2(\theta) + \sin^2(\theta) = 1\). Students reason about why the equation is an identity by drawing in right triangles for specific cases and then generalizing for any angle, using the equation of the unit circle (MP8).

The notation for squaring cosine and sine is introduced in this lesson as \(\cos^2(\theta)\) and \(\sin^2(\theta)\) so students do not, for example, confuse \(\sin^2(\theta)\) with \(\sin(\theta^2)\).

Learning Goals

Teacher Facing

  • Comprehend that cosine and sine are defined as the $x$- and $y$-coordinates of a point $P$ on the unit circle at $\theta$ radians.
  • Understand the Pythagorean Identity as a connection between the Pythagorean Theorem and the coordinates of points on a unit circle.
  • Use the Pythagorean Identity to determine if a point is on the unit circle.

Student Facing

  • Let’s learn more about cosine and sine.

Required Materials

Learning Targets

Student Facing

  • I can use the Pythagorean Identity to calculate values of coordinates given one coordinate to start from.
  • I understand that the coordinates of a point on the unit circle at $\theta$ radians can be written as $(\cos(\theta),\sin(\theta))$.

CCSS Standards

Building On

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.