6.2: Sine. It’s the Law.
This activity is intended to supplement Trigonometry, Chapter 5, Lesson 3.
ID: 11851
Time Required: 15 minutes
Activity Overview
In this activity, students will explore the Law of Sines. Students will derive the formula through exploration and solve some application problems. As an extension, students will prove the Law of Sines through guided questions.
Topic: Right Triangles & Trigonometric Ratios
- Law of Sines
Teacher Preparation and Notes
- This activity is geared towards geometry students and only the simplest case of the Law of Sines is explored. The ambiguous case is not explored in this activity.
- This activity was written to be explored with the Cabri Jr. app on the TI-84.
- 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=11851 and select LAW1.
Associated Materials
- Student Worksheet: Sine. It's the Law. http://www.ck12.org/flexr/chapter/9703, scroll down to the second activity.
- Cabri Jr. Application
- LAW1.8xv
Problem 1 – Law of Sines
Students will begin this activity by looking at a triangle and investigating the ratio of the sine of an angle to the length of the opposite side. In LAW1.8xv, students are given triangle \begin{align*}ABC\end{align*} with the measures of angles \begin{align*}A, B,\end{align*} and \begin{align*}C\end{align*}, and the measure of sides \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*}.
Students will collect data in the tables on their accompanying worksheet and asked what they notice about the last three columns of the table in Question 2. Discuss round off errors with the students.
Students are asked to compare the three columns to discover the Law of Sines.
Problem 2 – Application of the Law of Sines
In Problem 2, students are asked to apply what they have learned about the Law of Sines.
Extension – Proof of the Law of Sines
As an extension, students are asked to prove the Pythagorean theorem through guided questions.
Solutions
1.
Position | \begin{align*}a\end{align*} | \begin{align*}b\end{align*} | \begin{align*}c\end{align*} | \begin{align*}A\end{align*} | \begin{align*}B\end{align*} | \begin{align*}C\end{align*} |
---|---|---|---|---|---|---|
1 | 5.88 | 8.00 | 5.12 | 47.05 | 93.35 | 39.60 |
2 | 8.97 | 8.00 | 4.01 | 89.99 | 63.44 | 26.57 |
3 | 8.16 | 8.00 | 3.01 | 81.90 | 76.71 | 21.39 |
4 | 4.84 | 8.00 | 4.00 | 27.41 | 130.24 | 22.35 |
2.
Position | \begin{align*}\sin (A)\end{align*} | \begin{align*}\sin (B)\end{align*} | \begin{align*}\sin (C)\end{align*} | \begin{align*}\frac{\sin (A)}{a}\end{align*} | \begin{align*}\frac{\sin (B)}{b}\end{align*} | \begin{align*}\frac{\sin (C)}{c}\end{align*} |
---|---|---|---|---|---|---|
1 | 0.73135 | 0.99829 | 0.63742 | 0.12347 | 0.12478 | 0.12449 |
2 | 1 | 0.89446 | 0.44729 | 0.11148 | 0.11180 | 0.11154 |
3 | 0.99002 | 0.97321 | 0.36471 | 0.12132 | 0.12165 | 0.12116 |
4 | 0.46035 | 0.76334 | 0.38026 | 0.09511 | 0.09541 | 0.09506 |
3. They are approximately equal (not exactly equal due to round off errors).
4. \begin{align*}\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\end{align*}
5. \begin{align*}\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\end{align*}
6. Distance from Tower \begin{align*}A: 2.45 \ mi\end{align*}; Distance from Tower \begin{align*}B: 3.99 \ mi\end{align*}
7. 36.95 ft tall
8. Distance from lighthouse \begin{align*}A: 6.65 \ mi\end{align*}; Distance from lighthouse \begin{align*}B: 9.38 \ mi\end{align*}
9. \begin{align*}\sin (A) = \frac{BD}{c}\end{align*}
10. \begin{align*}\sin (C)= \frac{BD}{a}\end{align*}
11. \begin{align*}BD; c \cdot \sin (A) = BD\end{align*} and \begin{align*}a \cdot \sin (C) = BD\end{align*}
12. \begin{align*}c \cdot \sin (A) &= a \cdot \sin (C)\\ \sin (A) &= \frac{a \cdot \sin (C)}{c}\\ \frac{\sin (A)}{a} &= \frac{\sin (C)}{c}\end{align*}