Unit 4, Lesson 8
Sine and Cosine in the Same Right Triangle
Geometry
8.1
Warm-up
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four triangles. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the triangles for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three triangles that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.
Activity
NoneWhich three go together? Why do they go together?
A
B
C
D
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
The goal of this discussion is to review students’ reasoning for why each group of three triangles goes together.
Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “complementary angles” and “congruent,” and to clarify their reasoning as needed. Consider asking:
- “How do you know ?”
- “What do you mean by ?”
- “Can you say that in another way?”
If not mentioned by students, remind the class that trigonometry applies only to right triangles.
Student Lesson Summary
In previous lessons, we recalled that any right triangle with acute angles of 25 and 65 degrees was similar to any other right triangle with these same acute angles. Revisiting these triangles, we notice that the sine of 25 degrees is equal to the cosine of 65 degrees, and the cosine of 25 degrees is equal to the sine of 65 degrees.
| cosine of angle | sine of angle | |
|---|---|---|
| angle | adjacent leg hypotenuse | opposite leg hypotenuse |
| 0.906 | 0.423 | |
| 0.423 | 0.906 |
Mathematicians often use Greek letters to represent angles. For instance, is a Greek letter we use frequently in trigonometry. Looking at a general right triangle, the angles can be written as 90, , and . Using this general right triangle, we can fill out the table again. Notice that the same fraction, , appears for both and . That’s how we know for any acute angle .
| cosine of angle | sine of angle | |
|---|---|---|
| angle | adjacent leg hypotenuse | opposite leg hypotenuse |