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.

If students are mixing up adjacent and opposite sides, such as writing  for the first question, consider asking:

• “Can you explain how you found your answer.”

• “Put your finger on angle . Which leg is opposite angle ? Which leg is adjacent to angle ?”

The purpose of this discussion is to highlight some of the things we can and cannot conclude from knowing the values of trigonometric ratios associated with triangles. After reviewing the ratios for triangle , select students to share their explanations for how they drew triangle . An important point here is that while we know that triangle is isosceles and that the tangent of angle is 1, we don’t know the actual size of the triangle without more information.

Conclude the discussion by displaying the image for triangle . Here are some questions for discussion:

• “How is the information given for triangle different from the information given for triangle ?” (Since we know the coordinates of the 3 vertices, we can calculate the unknown side lengths and figure out the trig ratios from there.)
• “Suppose that all you knew about triangle was that . What else could you figure out? What couldn’t you figure out?” (Since sine is a ratio, I could find possible values for the side lengths using the Pythagorean Theorem. I wouldn’t know the actual side values without getting more information.)

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.

Have students discuss with a partner how the coordinates of and are related. They should consider how each coordinate value is related to the others, then how each coordinate is related to the cosine or sine of the angle at the origin. After partner discussion time, select pairs to share their observations, and record these for all to see. If not brought up by students, highlight how the coordinates of are the cosine and sine of the angle at the origin since the length of the hypotenuse, , is 1.

Students will use this type of reasoning to determine the coordinates of a point on a unit circle in a future lesson, so there is no need to expand this thinking to other quadrants at this time.