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Sine and Cosine in the Same Right Triangle

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

CCSS Standards

Building On

  • HSG-SRT.C.6

Routines and Materials

Instructional Routines

  • Which One Doesn’t Belong?

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. 

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?

Expand image

Student Response

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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)

  • HSG-SRT.C.7
  • MLR8: Discussion Supports

Required Materials

  • Scientific calculators

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.

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.

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

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

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

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)

  • Tools for creating a visual display

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.

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.

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

Are you ready for more?

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

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.

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)

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:


  • \(\sin(\theta)=\cos(\theta)\)


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

Students should notice: 

  • \(\sin(\theta)<\cos(\theta)\)  if  \(\theta<45\)
  • \(\sin(\theta)=\cos(\theta)\)  if  \(\theta=45\)
  • \(\sin(\theta)>\cos(\theta)\)  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,  \(\sin(\theta)=\frac{\text{short leg}}{\text{hypotenuse}}\) ,  \(\cos(\theta)=\frac{\text{long leg}}{\text{hypotenuse}}\)

\(\sin(\theta)=\cos(\theta)\) ​​​​​​

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

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.

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.

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