2.1: Trigonometric Ratios
This activity is intended to supplement Trigonometry, Chapter 1, Lesson 3.
ID: 9534
Time required: 35 minutes
Activity Overview
Students will use a graphing calculator to discover the relationship between the trigonometric functions: sine, cosine and tangent and the side length ratios of a right triangle.
Topic: Trigonometric Functions
- Solve any right triangle given an angle and the length of an opposite or adjacent side.
- Use technology to obtain the sine, cosine, or tangent of any angle.
Teacher Preparation and Notes
Students will use the graphing calculator to discover the relationship between the trigonometric functions: sine, cosine and tangent and the side length ratios of a right triangle. Prior to beginning the activity, students should download the CabriJr file TRIG to their graphing calculators.
This activity is designed as an introduction to the world of trigonometry. Students will explore the trigonometric ratios (sine, cosine, tangent) of a right triangle.
- This activity requires students to drag a point in CabriJr. If students are not familiar with this function of the CabriJr application, extra time should be taken to explain this.
- This activity is intended to be teacher-led. You may use the following pages to present the material to the class and encourage discussion. Students will follow along using their handhelds, although the majority of the ideas and concepts are only presented in this document; be sure to cover all the material necessary for students’ total comprehension.
- The student worksheet is intended to guide students through the main ideas of the activity. It also serves as a place for students to record their answers. Alternatively, you may wish to have the class record their answers on separate sheets of paper, or just use the questions posed to engage a class discussion.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
- To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=9534 and select TRIG.
Associated Materials
- Student Worksheet: Trig Ratios http://www.ck12.org/flexr/chapter/9699
- Cabri Jr. Application
- TRIG.8xv
In this activity, students will explore:
- Sine, cosine and tangents of angles
- Side length ratios of a right triangle
- Finding the missing side of a triangle when an angle and side are given.
Note: Students will need to exit the application in order to find the decimal values for the ratios and the trigonometric values for angle A. One option is for students to go through the TRIG application first, fill in all the ratios on the worksheet, and copy down the measures of angle A for each triangle. Then they would go back and complete the rest of the worksheet. Alternatively, students could work in pairs with one using the application and the other performing the calculations.
Problem 1 – Trigonometric Ratios
Students will open the TRIG file using the CabriJr application. Given a triangle, students will find the relationship between the ratios of the side lengths and the trigonometric functions.
Record the following ratios and trigonometric values to two decimal places.
\begin{align*}\frac{BC}{AC} & = \frac{4.0}{5.9}, \frac{AC}{AB} = \frac{5.9}{7.1}, \frac{BC}{AB} = \frac{4.0}{7.1}\\ \\ \text{Sin} \ A & = \underline{0.56}, \text{Cos} \ A = \underline{0.83}, \text{Tan} \ A = \underline{0.67}\end{align*}
Note: Make sure all student calculators are in DEGREE mode.
Make sure students find the decimal values below and compare the answers with the trig values.
\begin{align*}\frac{BC}{AC} &= 0.67\\ \frac{AC}{AB} &= 0.83\\ \frac{BC}{AB} &= 0.56\end{align*}
Students are asked to repeat this process for two more different triangles by moving point B to a different location (to grab a point, press the ALPHA button and use the arrow keys to move it.)
Answers will vary. Students should check their answers with the trig values. Answers should match the trig identities.
Based upon your answers hypothesize which ratio goes with each trigonometric function.
\begin{align*}\text{Sin} \ A = \frac{BC}{AB} && \text{Cos} \ A = \frac{AC}{AB} && \text{Tan} \ A = \frac{BC}{AC}\end{align*}
A good acronym to use to help remember these relationships is SOHCAHTOA
\begin{align*}\sin A &=\frac{\text{Opposite}}{\text{Hypotenuse}}\\ \cos A &= \frac{\text{Adjacent}}{\text{Hypotenuse}}\\ \tan A &=\frac{\text{Opposite}}{\text{Adjacent}}\end{align*}
Problem 2 – Trigonometry, What Is It Good For?
One of the uses of trigonometry is finding missing side lengths of a triangle.
To find the length of side \begin{align*}BC\end{align*} in the triangle to the right, write the sine relationship.
\begin{align*}\sin27=\frac{BC}{24}\end{align*}
Now solve for \begin{align*}BC\end{align*} and calculate using your handheld.
\begin{align*}24\sin27 &= BC\\ 24 \cdot 0.45 &= BC\\ 10.90 &= BC\end{align*}
To find the length of side \begin{align*}AC\end{align*} in the triangle to the right, write the cosine relationship.
\begin{align*}\cos48 =\frac{AC}{19}\end{align*}
Now solve for \begin{align*}AC\end{align*} and calculate using your handheld.
\begin{align*}19\cos48 = AC\\ 19 \cdot 0.67 = AC\\ 12.71 = AC\end{align*}
To find the length of side \begin{align*}AC\end{align*} in the triangle to the right, write the tangent relationship.
\begin{align*}\tan50=\frac{10}{b}\end{align*}
Now solve for \begin{align*}AC\end{align*} and calculate using your handheld.
\begin{align*}b\tan50 &= 10\\ b = \frac{10}{\tan50} &= \frac{10}{1.19} = 8.4\end{align*}
Solutions
- \begin{align*}AC = 14.62\end{align*}
- \begin{align*}BC = 19.51\end{align*}
- \begin{align*}AC = 49.52\end{align*}
- \begin{align*}AC = 95.09\end{align*}