The purpose of this Warm-up is for students to recall that a point, , on the unit circle with angle in Quadrant I has coordinates . When students articulate their reasoning, they have an opportunity to attend to precision in the language that they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

Previously, the connection was made between the coordinates of a point 1 unit away from the origin in Quadrant I at angle and the trigonometric ratios cosine and sine. Specifically, the coordinates of the point are . The purpose of this Warm-up is for students to recall that connection in the context of the unit circle and a point, , in Quadrant I. Throughout the lesson, students will extend this connection to other points on the unit circle not in Quadrant I, leading up to establishing the Pythagorean Identity, .

Monitor for students who draw and label a right triangle with side lengths and , and ask them to share during the class discussion.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the exact coordinates of . In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner's ideas and writing.

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”
• “How do you know that is exactly ?

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words that they got from their partners, to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, select previously identified students to share their reasoning, including the connection that they make between side lengths and the value of the cosine and sine of the angle when the hypotenuse is 1 unit. Display their work for all to see. Use a scientific calculator set to radians, or tell students to use their own, to approximate the value of and , and see that these values match the approximations that students identified in an earlier lesson, . If time allows, tell students to check their earlier approximations of cosine and sine for -radian and -radian.

Conclude the discussion by applying the Pythagorean Theorem to the exact and approximate coordinates. On the image from the activity, draw in (for students to see) the right triangle for point . Label the side lengths. Using the approximate value of , we see that the value of is slightly larger than . Using the exact values, is exactly . Tell students that previously they identified approximate values for the coordinates of points on the unit circle. Today, they will learn how to find the exact values.

If students are unfamiliar with the notation for raising a trigonometric function to a power, tell them that we use and to represent the square of the cosine of an angle and the square of the sine of an angle , respectively. This avoids the possible confusion that the angle is being squared, which happens when writing the expression as , for example.

The goal of this activity is to redefine cosine and sine using the unit circle. Previously, students have used cosine and sine only in reference to right triangles. This is an important step as students transition to an understanding of cosine and sine as functions.

Monitor for students who:

• Create a right triangle in Quadrant II and reason about the coordinates of , using the angle  radians and what they know about point .
• Repeat the calculations from the Warm-up and identify as the coordinates for , possibly by checking the values against the approximations they made previously.

The purpose of this activity is for students to further their understanding of the relationship between cosine, sine, the Pythagorean Theorem, and the unit circle in order to show that the Pythagorean Identity is always true. Students study some specific points to determine if they are, or are not, on the unit circle, by reasoning about the structure of points on a circle (MP7). During the synthesis, the Pythagorean Identity is defined as , for any angle . Students continue to work with this identity in the following lesson.