Lesson 5

The Pythagorean Identity (Part 1)

5.1: Circle Equations (5 minutes)

Warm-up

Previously, the connection between the coordinates of a point 1 unit away from the origin in quadrant 1 at angle \(\theta\) and the trigonometric ratios cosine and sine was made. Specifically, the coordinates of the point are \((\cos(\theta),\sin(\theta))\). The purpose of this warm-up is for students to recall that connection in the context of the unit circle and a point \(P\) in quadrant 1. Students will extend this connection to other points on the unit circle not in quadrant 1 throughout the lesson, leading up to establishing the Pythagorean Identity, \(\cos^2(\theta)+\sin^2(\theta)=1\).

Monitor for students who draw and label a right triangle with side lengths \(\cos(\frac{\pi}{3})\) and \(\sin(\frac{\pi}{3})\) to share during the class discussion.

Student Facing

Here is a circle centered at \((0,0)\) with a radius of 1 unit.

What are the exact coordinates of \(P\) if \(P\) is rotated counterclockwise \(\frac{\pi}{3}\) radians from the point \((1,0)\)? Explain or show your reasoning.

A circle on a coordinate plane, center at the origin O, radius 1. Point P on circle in quadrant 1, central angle theta.

Student Response

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

Anticipated Misconceptions

While students have seen \(\theta\) used for angles in a previous course, they may not recall its name or think it has a special meaning or properties due to not being part of the regular English alphabet. Remind them that it is the Greek letter theta, and that it is often used in mathematics to denote an angle.

Activity Synthesis

Select previously identified students to share their reasoning, displaying their work for all to see as well as the connection between side lengths and the value of the cosine and sine of the angle when the hypotenuse is 1 unit. Use a scientific calculator set to radians, or tell students to use their own, to approximate the value of \(\cos(\frac{\pi}{3})\) and \(\sin(\frac{\pi}{3})\) and see that these values match the approximations students identified in an earlier lesson, \((0.5, 0.87)\). If time allows, tell students to check their earlier approximations of cosine and sine for \(\frac{\pi}{6}\) radians and \(\frac{\pi}{4}\) radians.

Conclude the discussion by applying the Pythagorean Theorem to the exact and approximate coordinates. Begin by drawing in the right triangle for point \(P\) for students to see on the image from the activity and labeling the side lengths. Using the approximate value of \((0.5,0.87)\), we see that the value of \(0.5^2+0.87^2\) is slightly larger than \(1^2\). Using the exact values, \(\cos^2\left(\frac{\pi}{3}\right)+\sin^2\left(\frac{\pi}{3}\right)\) is exactly \(1^2\). Tell students that previously they identified approximate values for the coordinates of points on the unit circle. Today, they will learn how to find the exact values.

If students are unfamiliar with the notation for raising a trigonometric function to a power, tell them that we use \(\cos^2(\theta)\) and \(\sin^2(\theta)\) to represent the square of the cosine of an angle \(a\) and the square of the sine of an angle \(\theta\), respectively. This avoids the possible confusion that the angle is being squared that happens when writing the expression as \(\cos(\theta)^2\), for example.

5.2: Cosine, Sine, and the Unit Circle (15 minutes)

Activity

The goal of this activity is to redefine cosine and sine using the unit circle. Previously, students have only used cosine and sine in reference to right triangles. This is an important step as students transition their understanding of cosine and sine as functions.

Monitor for students who:

  • create a right triangle in quadrant 2 and reason about the coordinates of \(Q\) using the angle \(\frac{2\pi}{3}\) radians and what they know about point \(P\)
  • repeat the calculations from the warm-up and identify \(\left(\cos\left(\frac{2\pi}{3}\right), \sin\left(\frac{2\pi}{3}\right)\right)\) as the coordinates for \(Q\), possibly by checking the values against the approximations they made previously
  • notice that \(\left(\text-\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right)\) is the same as \(\left(\cos\left(\frac{2\pi}{3}\right), \sin\left(\frac{2\pi}{3}\right)\right)\)

Launch

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software. 
Supports accessibility for: Organization; Conceptual processing; Attention

Student Facing

What are the exact coordinates of point \(Q\) if it is rotated \(\frac{2\pi}{3}\) radians counterclockwise from the point \((1,0)\)? Explain or show your reasoning.

A circle with center O at the origin of an x y plane.

 

Student Response

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

Anticipated Misconceptions

Watch for students getting incorrect values when they use technology to find the values of cosine and sine for different angles. In some cases, the technology may need to be changed from degrees to radians. In other cases, remind students to use parentheses when calculating values. For example, typing in \(\sin\left(\frac{11\pi}{12}\right)\) without the correct format can return a value of 0.

Activity Synthesis

Select previously identified students to share in the order listed in the Activity Narrative, highlighting any right triangles or added labels students used. An important connection for students to make here is that we can think about the coordinates of \(Q\) in different ways. We can always use a right triangle to reason about the coordinates of points on the unit circle, which in this case leads to the coordinates \(\left(\text-\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right)\) since the right triangle drawn in quadrant 2 has the angle \(\frac{\pi}{3}\) radians. Trying to work this way in the different quadrants leads to always having to use our understanding of the coordinate plane to make sure the correct sign is used for the coordinates. If instead we use the angle given, \(\frac{2\pi}{3}\) radians, the sign of the \(x\)-coordinate is correct because \(\cos\left(\frac{2\pi}{3}\right)\) is negative.

Using calculators, tell students to check at least two other points each in either quadrant 3 or quadrant 4 against the approximate coordinates they labeled on their unit circle from a previous lesson (or referencing one of the displayed unit circles). The purpose of checking these values using technology is twofold:

  • to convince students that the values of \(\cos(\theta)\) and \(\sin(\theta)\), where \(\theta\) is an angle greater than \(\frac{\pi}{2}\), have meaning in the context of a unit circle as the coordinates \((\cos(\theta),\sin(\theta))\) of the point located on the unit circle rotated \(\theta\) radians counterclockwise from the positive \(x\)-axis
  • to encourage students to use the angle \(\theta\) to identify coordinates instead of working out the right triangle in a particular quadrant and then having to consider the sign of the \(x\)- and \(y\)-values in that quadrant

Conclude the discussion by telling students that cosine and sine do exist for angles greater than \(\frac{\pi}{2}\) radians—a right angle. Specifically, the \(x\)-coordinate of a point on the unit circle rotated \(\theta\) radians counterclockwise from \((1,0)\) is \(\cos(\theta)\), while the \(y\)-coordinate is \(\sin(\theta)\).

Representing, Conversing: MLR7 Compare and Connect. Prior to the whole-class discussion, invite students to create a visual display of their work for finding the exact coordinates of point \(Q\) after the given rotation. Students should consider how to represent their strategy so that other students will be able to understand how they arrived at their solution. Students may wish to add notes or details to their displays to help communicate their reasoning and thinking. Begin the whole-class discussion by selecting and arranging 2–4 displays for all to see. Give students 1–2 minutes of quiet think time to interpret the displays before inviting the authors to share their strategies as described in the Activity Narrative. This will help students make connections between different ways to think about the coordinates of \(Q\) using cosine and sine on the unit circle.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

5.3: A New Identity (15 minutes)

Activity

The purpose of this activity is for students to further their understanding of the relationship between cosine, sine, the Pythagorean Theorem, and the unit circle in order to show that the Pythagorean Identity is always true. Students study some specific points to determine if they are, or are not, on the unit circle by reasoning about the structure of points on a circle (MP7). During the synthesis, the Pythagorean Identity is defined as \(\cos^2(\theta) + \sin^2(\theta)=1\) for any angle \(\theta\). Students continue to work with this identity in the following lesson.

Launch

Arrange students in groups of 2. Ask, “Is the point \(\left(\frac{\sqrt{2}}{2},\text- \frac{\sqrt{2}}{2}\right)\) on the unit circle? Be prepared to explain how you know.” (Yes, because the coordinates satisfy the equation for the unit circle, \(x^2+y^2=1\).) After some work time, select students to share their thinking. Record any equations students used to reach their conclusions, such as the equation for a circle of radius \(r\) or the Pythagorean Theorem. If both these equations were not discussed, display them now and invite students to explain how they could use the equation to show that the point is on the unit circle.

If students ask where the coordinates with the square roots came from, let them know that they are the exact coordinates for the point at \(\theta=\frac{7\pi}{4}\), which was approximated in an earlier lesson as \((0.71,\text-0.71)\) for \((\cos(\frac{7\pi}{4}), \sin(\frac{7\pi}{4}))\).

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. As students share their reasoning about how they determined if the point is on the unit circle, scribe their thinking on a visible display. For example, highlight the \(x\)-coordinate, \(x^2\), the horizontal leg of the right triangle and \(a^2\) in the Pythagorean Theorem the same color. 
Supports accessibility for: Visual-spatial processing; Conceptual processing

Student Facing

  1. Is the point \(\left(\text-0.5,\sin(\frac{4\pi}{3})\right)\) on the unit circle? Explain or show your reasoning.
  2. Is the point \(\left(\text-0.5,\sin(\frac{5\pi}{6})\right)\) on the unit circle? Explain or show your reasoning.
  3. Suppose that \(\sin(\theta)=\text-0.5\) and that \(\theta\) is in quadrant 4. What is the exact value of \(\cos(\theta)\)? Explain or show your reasoning.

Student Response

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

Student Facing

Are you ready for more?

Show that if \(\theta\) is an angle between 0 and \(2\pi\) and neither \(\cos(\theta)=0\) nor \(\sin(\theta)=0\), then it is impossible for the sum of \(\cos(\theta)\) and \(\sin(\theta)\) to be equal to 1.

Student Response

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

Activity Synthesis

The goal of this discussion is for students to conclude that the value of \(\cos^2(\theta) + \sin^2(\theta)\) is 1 for any angle \(\theta\) and reason about why this is true. Students are also introduced to the formal name for this equation as the Pythagorean Identity.

Ask, “If \(\theta\) is an angle on the unit circle, what is the value of \(\cos^2(\theta) + \sin^2(\theta)\)?” (All points \((x,y)\) on the unit circle make the equation of the circle, \(x^2+y^2=1^2\), true. So \(\cos^2(\theta) + \sin^2(\theta)=1\) since \(x=\cos(\theta)\) and \(y=\sin(\theta)\). Or, for a point \((x,y)\) on the unit circle, we can draw a right triangle with leg of length \(|\cos(\theta)|\) and \(|\sin(\theta)|\) and hypotenuse 1, so, by the Pythagorean Theorem, \(\cos^2(\theta) + \sin^2(\theta)=1\).) Give partners 2–3 minutes of work time and then pair groups together to share and refine their reasoning. Invite students to share their group’s thinking with the whole class, displaying any diagrams created for all to see. If both approaches—using the equation for the unit circle and using the Pythagorean Theorem—are not brought up, do so now.

Conclude the discussion by telling students that \(\cos^2(\theta) + \sin^2(\theta)=1\), where \(\theta\) is an angle on the unit circle, is known as the Pythagorean Identity (If necessary, remind students that identities are equations that are true for all values of the variable in them). In addition to determining if points are and are not on the unit circle, this identity can be used to, for example, calculate the value of the sine of an angle from the value of the cosine of the angle. Using the identity this way is the focus of the next lesson.

Lesson Synthesis

Lesson Synthesis

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 since either \(\cos(\theta)\) or \(\sin(\theta)\) is 0. Similar to other points on the unit circle, if all we know is either the \(x\)- or \(y\)-coordinate, then the Pythagorean Identity isn’t enough to determine the exact location of a point on the unit circle. This particular aspect of the identity is the focus of the next lesson.

Display a unit circle for all to see, such as the one given here:

Circle centered at origin 

Ask, “If \(\theta\) is an angle measure on the unit circle, and \(\cos(\theta)=0\) at a point \(T\) on the unit circle, what else do you know about point \(T\)?” (Point \(T\) is at either \((0,1)\) or \((0,\text-1)\).) Invite students to share their reasoning about the possible locations of point \(T\). Highlight how a right triangle cannot be drawn to reason about the location of point \(T\) since it is on the \(y\)-axis, but we can still use the Pythagorean Identity to reason about the value of \(\sin(\theta)\), the \(y\)-coordinate of \(T\).

An important takeaway here is that there are two inputs, whose angles are related by their distance from the \(x\)-axis, that have an output of 0 for cosine. This connects back to the repeating nature of the coordinates of points on the unit circle as we rotate around it.

5.4: Cool-down - The (Pythagorean) Identity of Any Point (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Let’s say we have a point \(P\) with coordinates \((a,b)\) on the unit circle, like the one shown here:

A circle on a coordinate plane, center at the origin, radius 1. Point P on circle with coordinates a comma b, central angle theta.

Using the Pythagorean Theorem, we know that \(a^2+b^2=1\). We also know this is true using the equation for a circle with radius 1 unit, \(x^2+y^2=1^2\), which is true for the point \((a,b)\) since it is on the circle.

Another way to write the coordinates of \(P\) is using the angle \(\theta\), which gives the location of \(P\) on the unit circle relative to the point \((1,0)\). Thinking of \(P\) this way, its coordinates are \((\cos(\theta),\sin(\theta))\). Since \(a=\cos(\theta)\) and \(b=\sin(\theta)\), we can return to the Pythagorean Theorem and say that \(\cos^2(\theta) + \sin^2(\theta) = 1\) is also true.

What if \(\theta\) were a different angle and \(P\) wasn’t in quadrant 1? It turns out that no matter the quadrant, the coordinates of any point on the unit circle given by an angle \(\theta\) are \((\cos(\theta),\sin(\theta))\). In fact, the definitions of \(\cos(\theta)\) and \(\sin(\theta)\) are the \(x\)- and \(y\)-coordinates of the point on the unit circle \(\theta\) radians counterclockwise from \((1,0)\). Up until today, we’ve only been using the quadrant 1 values for cosine and sine to find side lengths of right triangles, which are always positive.

This revised definition of cosine and sine means that \(\cos^2(\theta) + \sin^2(\theta) = 1\) is true for all values of \(\theta\) defined on the unit circle and is known as the Pythagorean Identity.