Lesson 2: Special Right Triangles
Standards Addressed
MCC9-12.F.TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.
MCC9-12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
MCC9-12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Activity Description
To complete this activity, students will need access to the online applet, paper, a pencil, and a calculator. The applet can be found on the "Applet Activity" subsection of this page, along with a set of instructions and questions for students to answer. The tools are easy to use, so teachers should not have to give many instructions before the work session. Students can work individually or in pairs, depending on the class size and resources available. On the activity page, students will find two applets, one for each special right triangle. After setting a length for the sides of their triangle, students will use the provided measures to discover the general ratios for each type of figure. There is also a prompt that asks them to consider how their conjectures could be proven. Finally, students will calculate the three basic trigonometric values for the angles in the triangles, leading to the creation of the unit circle in Lesson 4. After most of the class has reached the end of the activity, we recommend that the teacher facilitates a discussion to summarize the findings.
Activity Rationale
The primary object of this lesson is for students to discover the relationships between the side lengths of 45-45-90 and 30-60-90 triangles. This goal relates to those of the overall unit because the values discovered here help students transition to the unit circle later in the unit. After finding the trigonometric ratios for the angles in each triangle, students should be able to easily understand the origin of the numbers in the unit circle. In this activity, students will be reasoning both abstractly and quantitatively; they begin with numbers in two specific examples and move to considering general cases for all right triangles. Students then construct viable arguments in question four to explain why their conjecture is true. The tools presented in the applets must also be used strategically for students to reach the intended results. The technology corresponds with the learning objectives by allowing students to create many different examples of these triangles to explore their ideas. All of these possible cases will help students develop the relationships required in this lesson. These applets are also beneficial because the appropriate triangles are already constructed. Students are not able to quickly construct 45-45-90 or 30-60-90 triangles, unless the figures are already provided. In a lesson that did not use technology, students would not be able to easily manipulate the accurate figures to explore multiple examples.
Teacher Guidelines
We do not anticipate many student difficulties arising from this activity. The most likely one would occur with students' calculations. They could calculate the reverse ratios, such as AC to BC, to obtain the incorrect values. Teachers would need to monitor for this problem. One unfortunate consequence of calculating these ratios is that a calculator will round the answers. Thus, while the correct ratio is the square root of two, many students may think it is simply the number 1.41. Hopefully, many students will realize the actual value when proving their conjecture by the Pythagorean Theorem. This misconception could be addressed in the class discussion by asking students about the differences between rational and irrational numbers. The applet provides enough direction so that teachers will only need to explain how to access it. The only technical problem we can foresee is that students will not be able to see all of their triangle if they make the side lengths large. We addressed this issue in the prompts by explaining that the figures can be moved to have a better view. The following are possible questions to pose to students not included in the prompts:
- What else do you know about the sides of right triangles that could help you prove these relationships?
- What is the relationship between cos(60) and sin(30)? Why is this true?
- Do you think the ratio in the 45-45-90 triangle is exactly 1.41?
- If side AB in the 30-60-90 triangle is "x", how could you represent the other two sides in terms of that variable?
- How would you construct these triangles given a square and an equilateral triangle?
Assessment
A short quiz over the concepts learned in this lesson and the previous one can be found on Socrative. Teachers will be able to view live results from students in a variety of formats. The classroom code is "EMAT6700Trig". We have also included a short homework assignment for students to complete that lets them practice using the ratios they have discovered, as well as solutions to it.
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