Lesson 6

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.

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 \(\dfrac{\sin(\theta)}{\cos(\theta)}\).

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:

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.

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 \(\sin(\theta)=\text-0.96\)

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

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.

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).

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 \(\tan(\theta)\approx1.73\).

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.

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.)