Lesson 2

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.

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.

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.

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.

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. 

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 \(\frac34\).)

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.

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.)

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.

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.

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.