Unit 6, Lesson 2
Revisiting Right Triangles
Integrated Math 3
2.1
Warm-up
Instructional Routines
MLR5: Co-Craft QuestionsMaterials
NoneActivity Narrative
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.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the image of the right triangle and circle. Use Co-Craft Questions to orient students to the image and to elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation, before comparing questions with a partner.
Activity
NoneStudent Response
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Activity Synthesis
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.
2.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
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.
Lesson Synthesis
The goal of this discussion is for students to reflect on and make connections to what they already know about identifying coordinates, right triangles, trigonometry, and the Pythagorean Theorem.
Arrange students in groups of 2. Tell students they are going to consider a point, , in Quadrant I that is one unit away from the origin. Display an image of , like the one shown here, and remind students that for any possible point we can always draw in its vertical distance from the -axis and its distance to the origin in order to form a right triangle.
Display some or all of the list given here, and ask students, “How could you use each of the tools or information given to determine the actual coordinates of point ?”
- A ruler the same scale as the image
- The coordinates of the vertices of a triangle similar to triangle with a known hypotenuse length
- The length of either or
- The measure of angle (or angle )
After a brief quiet think time, invite students to explain how that information would help determine the coordinates of point . For example, given the measure of angle , we could calculate and , which gives the lengths of sides and , respectively.
Student Lesson Summary
Recall that the ratios of side lengths in similar right triangles are equivalent.
In this triangle, the cosine of angle is the ratio of the length of the side adjacent to angle to the length of the hypotenuse, or . The sine of angle is the ratio of the length of the side opposite angle to the length of the hypotenuse, or . The tangent of angle is the ratio of the length of the side opposite angle to the length of the side adjacent to angle , or .
Now consider triangle , which is similar to triangle with a hypotenuse of length 1 unit. Here is a picture of triangle on a coordinate grid:
Since the two triangles are similar, angles and are congruent. So how do the values of cosine, sine, and tangent of these angles compare to the angles in triangle ? It turns out that since all three values are ratios of side lengths, , , and .
Notice that the coordinates of are because segment has length and segment has length . In other words, the coordinates of are .