Lesson 8

Sine and Cosine in the Same Right Triangle

8.1: Which One Doesn’t Belong: Four Triangles (5 minutes)

Warm-up

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 then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Student Facing

Which one doesn’t belong?

A

Right triangle P Q R. Q R is x units, P Q is 5 units, angle P Q R is 70 degrees, and angle P R Q is 90 degrees.

B

Right triangle S T U. S U is y units, S T is 6 units, angle S T U is 20 degrees, and angle T S U is 90 degrees.

C

Triangle K L M. K M is x units, L M is 10 units, angle K M L is 115 degrees, and K L M is 15 degrees.

D

Right triangle G H J. G J is x units, G H is 2 point 5 units, and angle G H J is 90 degrees.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask each group to share one reason why a particular item does not belong. 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 one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use. Also, press students to defend unsubstantiated claims.

If not mentioned by students, remind the class that trigonometry only works with right triangles.

8.2: Twin Triangles (15 minutes)

Activity

In this partner activity, students take turns writing and solving trigonometric equations. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). This allows students to continue to practice applying trigonometry while simultaneously generating data for a compare/contrast activity. By comparing the two columns, students will notice the triangles are congruent since the angles given are complementary. The discussion will culminate in a conjecture that \(\sin(\theta)=\cos(90 - \theta)\) which students will prove in the next activity.

Launch

Arrange students in pairs or groups of 4. Tell half the group to work on column A while the other half of the group works on column B.

Conversing: MLR8 Discussion Supports. Use this routine to help students explain their strategy for writing and solving trigonometric equations. In their groups of 4, invite the students who solved problems in column A to begin with this sentence frame: “The value of _____ is _____, because _____.” Invite the listeners, the students who solved problems in column B, to press for additional details referring to specific features of the triangles and how they applied trigonometry. Students should switch roles for each problem. This will help students justify their strategies for solving trigonometric equations and notice the relationship that \(\sin(\theta)=\cos(90-\theta)\).
Design Principle(s): Support sense-making; Cultivate conversation
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite pairs to agree to solve 2 of the 3 problems in the column assigned. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

Your teacher will assign you to either Column A or Column B. Find the value of the variable for the problems in your column.

Column A:

Column B:

A1

Right triangle. Base is 7 units, hypotenuse is x units. Angle opposite the height is 41 degrees.

B1

Right triangle. Height of 7 units, hypotenuse of x, and angle opposite the height is 49 degrees.

A2

Right triangle. Base is y units, hypotenuse is 6 units. Angle opposite the base is 65 degrees.

B2

Right triangle. Base is y units, hypotenuse is 6 units. Angle opposite the height is 25 degrees.

A3

Right triangle. Height is 8 units, hypotenuse is z units. Angle opposite the height is 50 degrees.

B3

Right triangle. Base is 8 units, hypotenuse is z units. Angle opposite the height is 40 degrees.

Compare your solutions with your group's solutions. Why did you get the same answers to different problems?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share any equations they used and record each response for all to see.

Column A

Column B

A1: \(\cos(41)=\frac{7}{x}\)

B1: \(\sin(49)=\frac{7}{x}\)

A2: \(\sin(65)=\frac{y}{6}\)

B2: \(\cos(25)=\frac{y}{6}\)

A3: \(\sin(50)=\frac{8}{z}\)

B3: \(\cos(40)=\frac{8}{z}\)

Ask students to share the reasons they came up with for all the answers being the same. Discuss that each row is the same triangle because all the triangles have one right angle and the other two angles are complementary. So each row of equations is saying the same thing but from different perspectives. Cosine and sine have a special relationship, one students already explored before they knew the names cosine or sine. Ask students to think back to any conjectures they made while building the right triangle table. Discuss until students agree on a precise conjecture such as, “The sine of any acute angle is equal to the cosine of the complementary angle.”

“Mathematicians often use Greek letters to represent angles. Theta, \(\theta\), is a Greek letter we use frequently in trigonometry. What other Greek letters do you know?” (\(\pi, \alpha\) . . .)

“Let’s write the conjecture you just came up with using theta: \(\sin(\theta)=\cos(90 - \theta)\). In the next activity you will prove this conjecture.”

8.3: Explain the Co-nnection (15 minutes)

Activity

Students will take the examples from the previous activity and generalize them into a proof of \(\sin(\theta)=\cos(90 - \theta)\). Monitor for groups that create displays that communicate their mathematical thinking clearly, contain an error that would be instructive to discuss, or organize the information in a way that is useful for all to see.

During the lesson synthesis students will use this equation and generate more examples to explore the relationship between cosine and sine further by determining which angles cause sine to be less than cosine, equal to cosine, or greater than cosine.

Launch

Arrange students in groups of 2–4. Provide each group with tools for creating a visual display. Clarify that they will be writing a draft individually first, before they work as a group to write a convincing argument.

Action and Expression: Develop Expression and Communication. Provide options for communicating understanding. Invite students to create a bulleted list of their process with references to the labeled diagram.
Supports accessibility for: Language; Organization.

Student Facing

  1. Draw a diagram that will help you explain why \(\sin(\theta)=\cos(90 - \theta)\).
  2. Explain why \(\sin(\theta)=\cos(90 - \theta)\).

Discuss your thinking with your group. If you disagree, work to reach an agreement.

Create a visual display that includes:

  • A clearly-labeled diagram.
  • An explanation using precise language.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Make a conjecture about the relationship between \(\tan(\theta)\) and \(\tan(90-\theta)\).
  2. Prove your conjecture.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students struggle prompt them to draw a right triangle and label one of the acute angles \(\theta\). Ask them what is the measure of the other acute angle. (\(90-\theta\)) Then prompt students to label the sides with any variables.

Activity Synthesis

Select groups to share their visual displays. Encourage students to ask questions about the mathematical thinking or design approach that went into creating the display. Here are questions for discussion, if not already mentioned by students:

  • How can you clearly connect the explanation to the diagram? (label the parts, draw arrows, use phrases such as 'adjacent leg')
  • What type of triangle does this equation work for? (only right triangles)
Speaking, Representing: MLR8 Discussion Supports. Give students additional time to make sure that everyone in their group can explain their visual display and the relationships between the quantities represented. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

Give students a few minutes to respond to the prompt: “Use the visual displays and any other resources available to find values of \(\theta\) where:

  1. \(\sin(\theta)<\cos(\theta)\)
  2. \(\sin(\theta)=\cos(\theta)\)
  3. \(\sin(\theta)>\cos(\theta)\)

How does this relate to the equation \(\sin(\theta)=\cos(90 - \theta)\)?"

Students should notice: 

  1. \(\sin(\theta)<\cos(\theta)\) if \(\theta<45\)
  2. \(\sin(\theta)=\cos(\theta)\) if \(\theta=45\)
  3. \(\sin(\theta)>\cos(\theta)\) if \(\theta>45\)

Invite students to share their thinking. (We know \(\sin(\theta)=\cos(90 - \theta)\) so if we also want \(\sin(\theta)=\cos(\theta)\) that means \(\theta=90-\theta\). The only solution to \(\theta=90-\theta\) is 45 degrees. That's because 45 degrees is complementary to itself.)

Display sketches of triangles to solidify that, for example, \(\sin(\theta)<\cos(\theta)\) when \(\theta<45\) because when the angle is small the opposite leg will be shorter than the adjacent leg. So, \(\sin(\theta)=\frac{\text{short leg}}{\text{hypotenuse}}\)\(\cos(\theta)=\frac{\text{long leg}}{\text{hypotenuse}}\)

A right triangle.

\(\sin(\theta)<\cos(\theta)\)

A 45-45-90 triangle.

\(\sin(\theta)=\cos(\theta)\)​​​​​​

A right triangle.

\(\sin(\theta)>\cos(\theta)\)

8.4: Cool-down - Cosine’s Complement (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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.

Right triangle. Base labeled b, height labeled a, hypotenuse labeled c. Angle opposite side a is 25 degrees and angle opposite side b is 65 degrees.
angle cosine of angle = adjacent leg \(\div\) hypotenuse sine of angle = opposite leg \(\div\) hypotenuse
\(25^\circ\) 0.906 0.423
\(65^\circ\) 0.423 0.906

Looking at a general right triangle, the angles can be written as 90, \(\theta\), and \(90-\theta\). Mathematicians often use Greek letters to represent angles. For instance, \(\theta\) is a Greek letter we use frequently in trigonometry.

Right triangle A C B. Right angle at C. A is theta degrees. B is 90 minus theta degrees. A C is x. C B is y. B A is h.
  cosine of angle sine of angle
angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse
\(\theta^\circ\) \(\frac{x}{h}\) \(\frac{y}{h}\)
\((90-\theta)^\circ\) \(\frac{y}{h}\) \(\frac{x}{h}\)