Lesson 2

Revisiting Right Triangles

2.1: Notice and Wonder: A Right Triangle (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that we can use the trigonometric ratios learned in an earlier course to find coordinates of points on a circle, which will be useful when students are asked to identify specific points given only an angle and hypotenuse in a later activity. While students may notice and wonder many things about this image, recalling the relationships between the sides and angles of a right triangle are the important discussion points.

Launch

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

Circle, A. Triangle with vertex A, vertex B on the circle, vertex C inside the circle. Angle C is a right angle. Corresponding pposite sides of the triangle are labeled with lower-case a, b c.

 

Student Response

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

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If trigonometric ratios, such as cosine, sine, and tangent, do not come up during the conversation, ask students to discuss this idea. The ratios associated with the cosine, sine, and tangent of angle \(A\) (\(\frac{b}{c}\)\(\frac{a}{c}\), and \(\frac{a}{b}\), respectively) and identifying the values of these ratios are the focus of the next activity, so they do not need to go into that level of detail at this time.

2.2: Recalling Right Triangle Trigonometry (15 minutes)

Activity

The goal of this activity is to review the value of the cosine, sine, and tangent of an angle in a right triangle. Students consider different right triangles, their associated trigonometric ratios, and, in the last question, tie this work back to coordinates of points on a circle centered at \((0,0)\) from the previous lesson.

Depending on how much students remember from previous courses about trigonometric ratios and right triangles, familiarity with identifying the values of cosine, sine, and tangent of an angle will vary. Students will have more practice working with these ratios over the next several lessons.

Launch

Display for all to see the right triangle \(ABC\) shown here. The purpose of this launch is to help students recall the definitions of the cosine, sine, and tangent of an angle. 

Right triangle ABC 

Here are some questions for discussion. Record responses next to triangle \(ABC\) for students to reference throughout the activity.

  • “Which side is the hypotenuse of triangle \(ABC\) and what is its length?” (Side \(AB\) is opposite the right angle, so it is the hypotenuse of the triangle. Side \(AB\) has a length of 5.)
  • “Which side is adjacent to angle \(A\) and what is its length?” (Side \(AC\) is adjacent to angle \(A\) and has a length of 4.)
  • “What is the cosine of angle \(A\)?” (The ratio of side \(AC\) to side \(AB\) or \(\frac{4}{5}\).)
  • “What is the sine of angle \(A\)?” (The ratio of side \(BC\) to side \(AB\) or \(\frac{3}{5}\).)
  • “What is the tangent of angle \(A\)?” (The ratio of side \(BC\) to side \(AC\) or \(\frac34\).)
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code the hypotenuse, the side adjacent to angle \(A\), and the side opposite angle \(A\). As students share their reasoning about the ratios in right triangles, scribe their thinking on a visible display using the color coding to connect the parts of the ratio with the parts of the triangle.
Supports accessibility for: Visual-spatial processing; Conceptual processing

Student Facing

  1. Find \(\cos(A)\), \(\sin(A)\), and \(\tan(A)\) for triangle \(ABC\).
    Triangle A, B C. Angle B is a right angle. Side A, B length 15, side C B length 8, side A, C length 17.
  2. Sketch a triangle \(DEF\) where \(\sin(D)=\cos(D)\) and \(E\) is a right angle. What is the value of \(\tan(D)\) for this triangle? Explain how you know.
  3. If the coordinates of point \(I\) are \((9,12)\), what is the value of \(\cos(G)\), \(\sin(G)\), and \(\tan(G)\) for triangle \(GHI\)? Explain or show your reasoning.
    A circle on centered on a coordinate plane. Origin is labeled G = 0 comma 0. Point H at 9 comma 0. Point I lies  vertically above H and is on the circle. Triangle G H I is drawn with H a right angle.

Student Response

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

Anticipated Misconceptions

The changing orientation and labeling of the vertices of the right triangles from the example in the launch to the first question may confuse some students. Encourage these students to focus on the angle and identify which side is opposite the angle, which side is adjacent, and which is the hypotenuse. Some students may benefit from putting their finger on the angle as a reminder that the side their finger isn’t touching is the opposite side.

Activity Synthesis

The purpose of this discussion is to highlight some of the things we can and cannot conclude from knowing the values of trigonometric ratios associated with triangles. After reviewing the ratios for triangle \(ABC\), select students to share their explanations for how they drew triangle \(DEF\). An important point here is that while we know triangle \(DEF\) is isosceles and that the tangent of angle \(A\) is 1, we don’t know the actual size of the triangle without more information.

Conclude the discussion by displaying the image for triangle \(GHI\). Here are some questions for discussion:

  • “How is the information given for triangle \(GHI\) different from the information given for triangle \(DEF\)?” (Since we know the coordinates of the 3 vertices, we can calculate the unknown side lengths and figure out the trig ratios from there.)
  • “Suppose that all you knew about triangle \(GHI\) was that \(\sin(G)=\frac{12}{15}\). What else could you figure out? What couldn’t you figure out?” (Since sine is a ratio, I could find possible values for the side lengths using the Pythagorean Theorem. I wouldn’t know the actual side values without getting more information.)

2.3: Shrinking Triangles (15 minutes)

Activity

Students continue to consider the cosine, sine, and tangent of an angle in a right triangle in this activity, but now all triangles are on a coordinate grid with one vertex at the origin and one vertex of the triangle on the positive \(x\)-axis. Two important features of trigonometric ratios are highlighted. First, similar right triangles have the same side ratios. Second, the case where the hypotenuse has length 1 is of particular interest.

Right triangle ABC

In this case, the coordinates of \(B\) are \((\cos(A),\sin(A))\) because \(B\) is \(\cos(A)\) units over and \(\sin(A)\) units up from \((0,0)\). This important fact will be central to making sense of and studying the properties of \(\cos(A)\) and \(\sin(A)\) throughout this unit and for the future development of the unit circle.

Launch

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students to identify the side opposite angle \(D\), the side adjacent to angle \(D\), and the hypotenuse.
Supports accessibility for: Memory; Organization

Student Facing

  1. What are \(\cos(D)\), \(\sin(D)\), and \(\tan(D)\)? Explain how you know.

    Coordinate plane, x 0 to 12 by 2, y, 0 to 6 by 2. Triangle drawn from D at 0 comma 0 to E at 12 comma 0 to F at 12 comma 5. Angle E is a right angle.
  2. Here is a triangle similar to triangle \(DEF\).

    Coordinate plane, x 0 to 36 by 3, y, 0 to 18 by 3. Triangle drawn from D prime at 0 comma 0 to E prime at 36 comma 0 to F prime at 36 comma 15. Angle E prime is a right angle.
    1. What is the scale factor from \(\triangle DEF\) to \(\triangle D'E'F'\)? Explain how you know.
    2. What are \(\cos(D')\), \(\sin(D')\), and \(\tan(D')\)?
  3. Here is another triangle similar to triangle \( DEF\).

    Coordinate plane, x, 0 to 1 by point 5, y, 0 to 1 by point 5. A triangle, one vertex on the origin, one vertex on x axis between point 5 & 1, and the third vertex directly above. Hypotenuse length 1.
    1. Label the triangle \(D’'E’'F’'\).
    2. What is the scale factor from triangle \(DEF\) to triangle \(D’'E’'F’'\)?
    3. What are the coordinates of \(F’'\)? Explain how you know.
    4. What are \(\cos(D'')\), \(\sin(D'')\), and \(\tan(D'')\)?

Student Response

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

Student Facing

Are you ready for more?

Angles \(C\) and \(C’\) in triangles \(ABC\) and \(A’B’C’\) are right angles. If \(\sin(A) = \sin(A’)\), is that sufficient to show that \(\triangle ABC\) is similar to \(\triangle A’B’C’\)? Explain your reasoning.

Student Response

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

Activity Synthesis

Have students discuss with a partner how the coordinates of \(F\) and \(F''\) are related. They should consider how each coordinate value is related to each other, then how each coordinate is related to the cosine or sine of the angle at the origin. After partner discussion time, select pairs to share their observations, recording these for all to see. If not brought up by students, highlight how the coordinates of \(F''\) are the cosine and sine for the angle at the origin since the length of the hypotenuse, \(D''F''\), is 1.

Students will use this type of reasoning to determine the coordinates of a point on a unit circle in a future lesson, so there is no need to expand this thinking to other quadrants at this time.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-group discussion. After groups have discussed how the coordinates \(F\) and \(F’’\) are related, give students 2–3 minutes to plan what they will say when they present their responses with the whole class. Encourage students to consider what details are important to share, and to think about how they will explain their reasoning using mathematical language. 
Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

The goal of this discussion is for students to reflect and make connections to what they already know about identifying \((x,y)\) coordinates, right triangles, trigonometry, and the Pythagorean Theorem.

Arrange students in groups of 2. Tell students they are going to consider a point \(B\) in quadrant 1 that is 1 unit away from the origin. Display an image of \(B\), like the one shown here, and remind students that for any possible point \(B\) we can always draw in its vertical distance from the \(x\)-axis and its distance to the origin in order to form a right triangle.

Right triangle ABC

Display some or all of the list given here and ask students, “How could you use each of the tools or information given to determine the actual \((x,y)\) coordinates of point \(B\)?”

  • a ruler the same scale as the image
  • the coordinates of the vertices of a triangle similar to triangle \(ABC\) with a known hypotenuse length
  • the length of either \(BC\) or \(AC\)
  • the measure of angle \(A\) (or angle \(B\))

After a brief quiet think time, invite students to explain how that information would help determine the coordinates of point \(B\). For example, given the measure of angle \(A\), we could calculate \(\sin(A)\) and \(\cos(A)\), which gives the lengths of sides \(BC\) and \(AC\), respectively.

2.4: Cool-down - From Coordinates to Cosine (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

In an earlier course, we studied ratios of side lengths in right triangles.

Triangle A, B C. C is a right angle. Side A, C length 4, side B C length 3, side A, B length 5.

In this triangle, the cosine of angle \(A\) is the ratio of the length of the side adjacent to angle \(A\) to the length of the hypotenuse—that is \(\cos(A) = \frac{4}{5}\). The sine of angle \(A\) is the ratio of the length of the side opposite angle \(A\) to the length of the hypotenuse—that is \(\sin(A) = \frac{3}{5}\). The tangent of angle \(A\) is the ratio of the length of the side opposite angle \(A\) to the length of the side adjacent to angle \(A\)—that is \(\tan(A) = \frac{3}{4}\).

Now consider triangle \(A’B’C’\), which is similar to triangle \(ABC\) with a hypotenuse of length 1 unit. Here is a picture of triangle \(A’B’C’\) on a coordinate grid:

Coordinate plane, x, 0 to 1 by point 5, y, 0 to 1 by point 5. Triangle, A, prime at the origin, C prime on the x axis between point 5 and 1, B prime above C prime with y value between point 5 and 1.

Since the two triangles are similar, angle \(A\) and \(A'\) are congruent. So how do the values of cosine, sine, and tangent of these angles compare to the angles in triangle \(ABC\)? It turns out that since all three values are ratios of side lengths, \(\cos(A)=\cos(A')\), \(\sin(A)=\sin(A')\), and \(\tan(A)=\tan(A')\).

Notice that the coordinates of \(B’\) are \(\left(\frac{4}{5},\frac{3}{5}\right)\) because segment \(A’C’\) has length \(\frac{4}{5}\) and segment \(B’C’\) has length \(\frac{3}{5}\). In other words, the coordinates of \(B’\) are \((\cos(A'),\sin(A'))\).