In this lesson, students prove the Pythagorean Identity: . Students begin by collecting data on the squared values of cosine, sine, and tangent. Using their observations, students write conjectures about the data. They are directed to focus on for this lesson. Students then prove the identity and apply it to other problems.

The Pythagorean Identity is useful for advanced mathematics. It is also useful for students at this point in the unit to find the value of either cosine or sine, given only the value of the other trigonometric function, since students do not yet have access to inverse trigonometric functions on a calculator.

Students develop their proof of the Pythagorean Identity in stages. They brainstorm with a group, share plans with the class, write a draft in their group, and then give and receive more feedback using the Stronger and Clearer routine. They have repeated opportunities to explain their reasoning and critique the reasoning of others (MP3).

The following activity allows students to apply the identity and make more observations about the relationships among cosine, sine, and . Students may choose to use their Right Triangle Table during this activity. 

Note: Since, at this point in their learning, students understand trigonometry only in the context of right triangles, they will assume that all angles are acute (in quadrant one of the unit circle). While it’s true that if , could be either or , there is no need to address that complexity in this course.