The purpose of this Warm-up is to elicit the idea that we can use trigonometric ratios to find coordinates of points on a circle, which will be useful when students are asked to identify specific points given only an angle and hypotenuse in a later activity. While students may generate many mathematical questions about this image, recalling the relationships between the sides and angles of a right triangle are the important discussion points.

Invite several partners to share one question with the class, and record their responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as "coordinate," "hypotenuse," and "Pythagorean Theorem."

If no students ask about the angle measures of or , invite students to discuss this idea. 

The ratios associated with the cosine, sine, and tangent of angle (, , and , respectively) and identifying the values of these ratios are the focus of the next activity, so students do not need to go into that level of detail at this time.

The goal of this activity is to review the value of the cosine, sine, and tangent of an angle in a right triangle. Students consider different right triangles and their associated trigonometric ratios. In the last question, students tie this work back to coordinates of points on a circle that is centered at , from the previous lesson.

Depending on how much students remember from previous courses about trigonometric ratios and right triangles, familiarity with identifying the values of cosine, sine, and tangent of an angle will vary. Students will have more practice working with these ratios over the next several lessons.

Students continue to consider the cosine, sine, and tangent of an angle in a right triangle in this activity, but now all triangles are on a coordinate grid with one vertex at the origin and one vertex of the triangle on the positive -axis. Two important features of trigonometric ratios are highlighted. First, similar right triangles have the same side ratios. Second, the case where the hypotenuse has length 1 is of particular interest. As students study triangles with these features, they are making use of structure to explore trigonometric ratios (MP7).

In this case, the coordinates of are because is units over and units up from . This important fact will be central to making sense of and studying the properties of and throughout this unit and for the future development of the unit circle.