Lesson 1: Review of Right Triangle Trigonometry
Standard Addressed
MCC9-12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Activity Description
In this lesson students will review the three basic trigonometric functions, sine, cosine, and tangent by developing their own definitions of the functions using an online applet. The applet can be found in the subsection of this page labeled "Applet Activity." Included in this applet is a set of instructions and questions that students should answer individually. The students will need access to this online applet, paper, pencil, and a calculator. Once students have accessed the applet they will begin their exploration into the three trigonometric functions. This applet allows students to manipulate the angle measure of a right triangle to define the three basic trigonometric functions. Students are prompted to figure out how these three functions are defined before actually knowing the definitions of the functions. Students will achieve this by moving the green slider to adjust the angle measure while the two side lengths of the right triangle adjusts accordingly, the hypotenuse remains at 10. As the slider moves, the values for the three functions change as well. Students are prompted to use a calculator to help record their observations of how these three functions are related to the sides of the triangle. Furthermore, students are able to move point D at the bottom of the applet to change the lengths of the sides of the triangle, leaving the angle measure fixed, to demonstrate how these three ratios are independent of the triangles size. This applet should be used as a review of the trigonometric functions for students and once a majority of the class has finished their exploration, we recommend that the teacher engages in a discussion with the class about their definitions of the three functions.
Activity Rationale
This activity aims to provide students with a review of the three trigonometric functions; sine, cosine, and tangent by having them use the applet to develop their own definitions of the functions. This applet is connected to the unit in that it will allow students to make connections between these three values in special right triangles, and eventually in the Unit Circle at the end of the unit. This activity allows students to utilize their abstract and quantitative reasoning. Students must use the given side lengths to figure out how to receive the given value for each function. Furthermore, after they figure this out they must generalize their calculations so that they can apply them to any acute angle in a right triangle. This technology corresponds to the learning objectives in that students are able to move the angle measure by one degree to explore how the angle measure and side lengths are related to the three given values of the trigonometric functions. Furthermore, this technology prompts students to create their own definitions of the ratios without relying on SOHCAHTOA. This technology is beneficial to students because they will be able to work examples for angles measuring from one degree to eighty-nine degrees, providing them with plenty of practice finding the values for sine, cosine, and tangent.
Teacher Guidelines
For this activity the teacher is encouraged to be a facilitator to the students, answering questions as they arrive, while avoiding lecturing to the students. The only materials needed for this activity are internet access, pencil, paper, and a calculator. Many students may have difficulty at first reasoning how the values of the three trigonometric functions are achieved from the specific side lengths of the triangle. The instructor should then encourage and direct students by asking them various guided questions like the ones included below. By asking students these questions, the instructor is leading them to make the realization that they are preforming division in order to solve for the three trigonometric functions while not giving them the answer directly.
The directions for how to use this technology are pretty straight forward. The directions are included in the questions for the applet activity and are easy to follow. Therefore, the teacher should only have to tell students how to access the applet and should not have to explain how the applet works. Furthermore, there should be no technical issues when students are engaging with the applet. The following are possible questions to pose to students that are not included in the activity, but they could provide deeper thinking and understanding:
The students' responses would be sequenced in the order in which they were presented in the applet.
The directions for how to use this technology are pretty straight forward. The directions are included in the questions for the applet activity and are easy to follow. Therefore, the teacher should only have to tell students how to access the applet and should not have to explain how the applet works. Furthermore, there should be no technical issues when students are engaging with the applet. The following are possible questions to pose to students that are not included in the activity, but they could provide deeper thinking and understanding:
- In order to get a value less than one what does the relationship need to be between the numerator and the denominator?
- In order to get a value greater than one what does the relationship need to be between the numerator and the denominator?
- What do you notice about the values if you change the sides but do not change the angle and the hypotenuse? Keep the hypotenuse at 10.
- Is there anything specific you notice about the ratios of a triangle with angles 30-60-90?
The students' responses would be sequenced in the order in which they were presented in the applet.
Assessment
A short homework assignment is included, as well as the answers, in which students use their recently developed definitions of sine, cosine, and tangent and practice applying them to specific triangles.
References
Sibol, S. (n.d.). JMAP HOME - Math Regents Exams Integrated Algebra, Geometry, Trigonometry worksheets
answers lesson plans ExamView resources. JMAP HOME - Math Regents Exams Integrated Algebra,
Geometry, Trigonometry worksheets answers lesson plans ExamView resources. Retrieved July 31, 2014, from
http://www.jmap.org/index.html
answers lesson plans ExamView resources. JMAP HOME - Math Regents Exams Integrated Algebra,
Geometry, Trigonometry worksheets answers lesson plans ExamView resources. Retrieved July 31, 2014, from
http://www.jmap.org/index.html