This Math Talk focuses on determining the sign of the cosine, sine, and tangent of an angle . It encourages students to think about positive and negative values based on the quadrant and to rely on the structure of the coordinate plane to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students need to find the values of cosine, sine, and tangent for an angle, given the values of cosine, sine, or tangent and the quadrant. To describe the trigonometric values as the angle changes quadrants, students need to look for and make use of structure (MP7).

Up until now, students’ understanding of tangent has been limited to either right triangles or as the value of sine divided by the value of cosine. While the latter of these works for angles beyond those found in right triangles, during the whole-class discussion, students learn another way to think about tangent, using the unit circle, expanding their understanding of tangent beyond the angles possible within a right triangle.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking,

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”

In particular, highlight any strategies in which students used a unit circle diagram or sketched their own unit circle showing an angle .

End the discussion by displaying this image:

Say, “We learned earlier that we can think of as the coordinate of a point on the unit circle rotated radians counterclockwise from the point . We also know that . Where can you see the value of on this diagram?” After some quiet think time, invite students to share their ideas. If not brought up by students, point out that in the context of the unit circle, is the same as saying for the line connecting with the point . This means that one way to think about on the unit circle is as the slope of that line. In Quadrants I and III, that line has a positive slope, and in Quadrants II and IV, that line has a negative slope. At the end of the lesson, students will explore the value of tangent at the points where the unit circle intersects the axes, so there is no need to do so now.

The purpose of this activity is for students to make sense of Andre’s reasoning about the values of sine and tangent associated with an angle on the unit circle knowing only the value of cosine and the quadrant of the angle. As students consider Andre's reasoning, they make sense of problems involving the Pythagorean Identity (MP1).

Monitor for students who:

• Use a sketch of the unit circle to reason about the value of sine and tangent.
• Write out the Pythagorean Identity calculations to check Andre’s statement that .

Students sort different equations and expressions during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and to make connections (MP7). As students work, encourage them to refine their descriptions of possible values of sine, cosine, or tangent of an unknown angle in each quadrant, using more precise language and mathematical terms (MP6).

Students take turns matching cards. One set of cards shows the value of sine, cosine, or tangent of an unknown angle. The other set of cards shows a quadrant number on the unit circle. Students then select one of their possible matches and calculate the values of the two other trigonometric ratios.