Lesson 6

The Pythagorean Identity (Part 2)

6.1: Math Talk: Which Quadrant? (10 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for identifying the values of cosine, sine, and tangent of an angle \(\theta\) as positive or negative based on what quadrant \(\theta\) is in. These understandings help students develop fluency and will be helpful later in this lesson when students need to find the values of cosine, sine, and tangent for an angle given the values of cosine, sine, or tangent and the quadrant.

Up until now, students’ understanding of tangent has been limited to either right triangles or as the value of sine divided by the value of cosine. While the latter of these works for angles beyond those found in right triangles, during the whole-class discussion, students learn another way to think about tangent using the unit circle (MP2), expanding their understanding of tangent beyond the angles possible within a right triangle.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

If needed, remind students at the start of the warm-up that one definition of tangent of an angle \(\theta\) is \(\dfrac{\sin(\theta)}{\cos(\theta)}\).

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

For an angle \(\theta\) in the quadrant indicated, use mental estimation to identify the values of \(\cos(\theta)\), \(\sin(\theta)\), and \(\tan(\theta)\) as either positive or negative.

Quadrant 1

Quadrant 2

Quadrant 3

Quadrant 4

Student Response

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

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking,

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

In particular, highlight any strategies in which students used a unit circle diagram or sketched their own unit circle showing an angle \(\theta\).

End the discussion by displaying this image:

Circle with angle theta labeled 

Say, “We learned earlier that we can think of \((\cos(\theta),\sin(\theta))\) as the \((x,y)\) coordinate of a point on the unit circle rotated \(\theta\) radians counterclockwise from the point \((1,0)\). We also know that \(\tan(\theta)=\dfrac{\sin(\theta)}{\cos(\theta)}\). Where can you see the value of \(\tan(\theta)\) on this diagram?” After some quiet think time, invite students to share their ideas. If not brought up by students, point out that in the context of the unit circle \(\dfrac{\sin(\theta)}{\cos(\theta)}\) is the same as saying \(\dfrac{\text{rise}}{\text{run}}\) for the line connecting \((0,0)\) with the point \((\cos(\theta),\sin(\theta))\). This means that one way to think about \(\tan(\theta)\) on the unit circle is as the slope of that line. In quadrants 1 and 3, that line has a positive slope and in quadrants 2 and 4, that line has a negative slope. Students will explore the value of tangent at the points where the unit circle intersects the axes at the end of the lesson, so there is no need to do so now.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

6.2: Andre's Calculations (10 minutes)

Activity

The purpose of this activity is for students to make sense of Andre’s reasoning about the values of sine and tangent associated with an angle on the unit circle knowing only the value of cosine and the quadrant of the angle. This activity is meant to prepare students for the following card sort activity in which they will make similar calculations (MP1).

Monitor for students who:

  • use a sketch of the unit circle to reason about the value of sine and tangent
  • write out the Pythagorean Identity calculations to check Andre’s statements that \(\sin(\theta)=\text-0.96\)

Launch

Provide access to calculators. Give students quiet work time and then time to share their work with a partner.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, Andre _____ because . . .”, “I noticed . . .”, “Why did Andre . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Suppose that the angle \(\theta\), in radians, is in quadrant 4 of the unit circle. If \(\cos(\theta)=0.28\), what are the values of \(\sin(\theta)\) and \(\tan(\theta)\)?

Andre uses the Pythagorean Identity and determines that the value of \(\sin(\theta)\) is -0.96. Using the values of sine and cosine, he then calculates the value of tangent:

\(\begin{align} \tan(\theta)&=\dfrac{\sin(\theta)}{\cos(\theta)} \\ &=\frac{\text-0.96}{0.28} \\ &\approx \text-3.43 \end{align}\)

Do you agree with Andre? Explain or show your reasoning.

Student Response

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

Activity Synthesis

Invite previously identified students to share their reasoning about Andre’s solution. If possible, display a student sketch for all to see; otherwise, display the unit circle image from the Student Response.

An important takeaway here is that Andre’s calculations are all correct and that making a quick sketch of the unit circle is a good strategy for identifying if certain trigonometric values are positive or negative. If we didn’t know the quadrant, then we would be left not knowing if the value of \(\sin(\theta)\) should be positive or negative during the calculation steps because cosine is positive both in quadrants 1 and 4. The squaring in the Pythagorean Identity can make us forget that cosine, sine, and tangent can each have a negative value depending on the quadrant the angle is in on the unit circle.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their reasoning about Andre’s solution, present an incorrect response and explanation. For example, “I used the Pythagorean Identity to determine the value of \(\sin(\theta)\) and I got positive 0.96. This would make \(\tan(\theta)\) positive, so I disagree with Andre.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, monitor for students who draw a sketch of the unit circle and use the correct sign of cosine and sine in quadrant 4. This helps students evaluate, and improve upon, the written mathematical arguments of others, as they clarify how to find the values of sine and tangent given cosine and quadrant on the unit circle.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

6.3: Card Sort: Where's the Point? (15 minutes)

Activity

The goal of this matching activity is for students to practice making calculations with the Pythagorean Identity and using the structure of the unit circle (MP7). Students take turns matching cards. One set of cards shows the value of sine, cosine, or tangent of an unknown angle. The other set of cards shows a quadrant number on the unit circle. Students then select one of their possible matches and calculate the values of the two other trigonometric ratios. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Launch

Arrange students in groups of 2. Demonstrate how to lay out all the cards face up with those showing cosine, sine, or tangent on one side and those showing a quadrant number on the other. Tell students that they are going to take turns matching cards from each side as either possible or impossible. For example,  \(\cos(\theta)=0.5\) and quadrant 2 is an impossible match because cosine only has negative values in that quadrant. The goal is for each partner to identify 4 matches total, 2 possible and 2 impossible, so there will be some some cards left at the end of the first part.

Once each partner has completed their 4 matches, they then choose 1 of their possible matches to calculate the values of the other 2 trigonometric ratios. For example, if a card with the value of \(\sin(\theta)\) is drawn, the student will need to calculate the values of \(\cos(\theta)\) and \(\tan(\theta)\). After finishing their calculations, partners should take turns explaining their reasoning. Demonstrate productive ways to agree or disagree, for example, by explaining your mathematical thinking using different representations or asking clarifying questions.

Provide each group a set of cut-up cards for matching. Depending on the level of challenge needed, this activity may be completed with or without a calculator and with or without a unit circle diagram. If additional challenge is desired, tell students to make sure all values are exact instead of approximate. For example, writing \(\tan(\theta)=\dfrac{\text-\frac{\sqrt{3}}{2}}{\text-\frac{1}{2}}=\sqrt{3}\) instead of \(\tan(\theta)\approx1.73\).

Conversing: MLR8 Discussion Supports. In their groups of 2, students should take turns explaining their reasoning for how they determined if the matched cards are possible or not possible on the unit circle. Display the following sentence frames for all to see: “This match is/is not possible because . . .”, “I used the unit circle by . . .”, or “I noticed _____ so I . . . .” Encourage students to ask clarifying questions and challenge each other when they disagree. This will help students use mathematical language involving the Pythagorean Identity and the structure of the unit circle in communicating with a partner. 
Design Principle(s): Support sense-making; Maximize meta-awareness
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite each partner to identify 2 matches, 1 possible and 1 impossible. 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 give you a set of cards that should be arranged face up with cards showing values for sine, cosine, and tangent on one side and cards showing quadrants on the other.

  1. Take turns with your partner matching pairs of cards. Identify 2 pairs that are possible on the unit circle and 2 pairs that are not possible, in any order.
    1. For each pair, explain to your partner how you know if the pair is or is not possible on the unit circle. Once a pair is identified, place the cards in front of you to use later.
    2. For each pair that your partner draws, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
  2. Once you and your partner have identified 4 pairs each, pick 1 of your possible matches and then calculate the values of the two missing trigonometric ratios for that match. When you finish, trade calculations with your partner and check each other's work. If you disagree, discuss your thinking and work to reach an agreement.

Student Response

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

Student Facing

Are you ready for more?

Suppose \(\tan(\theta)=2\) and the angle \(\theta\) is in quadrant 1. Find the values of \(\cos(\theta)\) and \(\sin(\theta)\).

Student Response

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

Anticipated Misconceptions

As they work their calculations, students may be unsure when to take the positive or the negative expression when identifying the value of sine or cosine. Remind these students to sketch the situation determined by the two cards if they have not already done so, and to use that sketch to determine if the value they are calculating is positive or negative. For example, if they are calculating a cosine value in quadrant 4, it must be a positive value.

Activity Synthesis

Much discussion takes place between partners. Once all groups have completed the calculations for each pair of cards, discuss the following:

  • “Which matches were tricky? Explain why.” (Both the tangent cards. For example, starting from \(\tan(\theta)=1\) means we don’t have an exact \(x\)- or \(y\)-coordinate. Instead, we reasoned that the value of sine and cosine must be the same, and in quadrant 1, that only happens when \(\theta=\frac{\pi}{4}\).)
  • “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?” (At first, I thought a match was possible, but then my partner sketched a unit circle to show that in quadrant 2, \(\sin(\theta)=\text-0.5\) can’t happen since all the \(y\)-coordinates in quadrant 2 are positive.)

Lesson Synthesis

Lesson Synthesis

Similar to the previous lesson, the goal of this discussion is to call student attention to the 4 points on the unit circle where we cannot draw in a right triangle. For this lesson, we will address what this means for the value of tangent at these 4 angles, which students will study in more depth in a future lesson when they work with the graph of the tangent function.

Display a unit circle for all to see, such as the one here, with 4 points marked where the circle intersects the axes.

Circle centered at origin 

Here are some questions for discussion:

  • “What is the value of tangent at \((0,1)\) and \((0,\text-1)\)?” (\(\tan(\theta)\) does not exist since a vertical line does not have a slope.)
  • “What is the value of tangent at \((1,0)\) and \((\text-1,0)\)?” (\(\tan(\theta)=0\) since the line between those points and the origin has a slope of 0.)
  • “Are there any other points on the unit circle where the value of tangent is either 0 or does not exist?” (None, because all other points on the unit circle have non-zero \(x\)- and \(y\)-coordinates.)

6.4: Cool-down - $\sin(\theta)$ and a Quadrant (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Say we know that \(\sin(\theta)=\text-\frac{\sqrt{2}}{2}\) and that \(\theta\) is an angle in quadrant 3. What can we say about the values of cosine and tangent at \(\theta\)?

Since we can think of \(\sin(\theta)\) as the \(y\)-coordinate of a point \(P\) in quadrant 3, let’s start with a sketch of the unit circle showing point \(P\).

Circle on the coordinate plane, center at the origin. Point P on the circle in the third quadrant, central angle counter-clockwise from positive x axis has measure theta.

The sketch helps us see that the \(x\)-coordinate, which is \(\cos(\theta)\), is also negative. Using the Pythagorean Identity, we can calculate the value of \(\cos(\theta)\):

\(\begin{align} \cos^2(\theta)+\sin^2(\theta)&=1^2 \\ \cos^2(\theta)+\left(\text-\frac{\sqrt{2}}{2}\right)^2 &= 1 \\ \cos^2(\theta)&=1-\left(\text-\frac{\sqrt{2}}{2}\right)^2 \\ \cos^2(\theta)&=1-\frac{2}{4} \\ \cos(\theta)&=\text-\sqrt{1-\frac{1}{2}} \\ \cos(\theta)&=\text-\sqrt{\frac{1}{2}} \end{align}\)

Now that we know the value of cosine, we can calculate the value of tangent with some division:

\(\tan(\theta)=\dfrac{\text-\frac{\sqrt{2}}{2}}{\text-\sqrt{\frac{1}{2}}}=\dfrac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1\)

We can use one piece of information and the structure of the unit circle to figure out a whole bunch more, similar to how we used the value of one side length, the hypotenuse, and the structure of right triangles in the past to figure out the other side length of the right triangle.