# The Law of Sines: More than Meets the Eye

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## Objective

SWBAT use the Law of Sines and find the area of non-right triangles.

#### Big Idea

Using the Law of Sines will throw your students for a loop - there are two possible answers!

## Launch and Explore

10 minutes

After working with the Law of Cosines yesterday, students will be expecting to work with the Law of Sines next. We are going to move on to this formula, but it will have a different dimension than when they used it in Geometry since now we have knowledge of the unit circle and some additional properties of the sine ratio. I am not going to derive this formula since my students have used it in their Geometry course.

I have my students begin by brainstorming with their table what they know about the Law of Sines. See if they can recall what the formula is for and what it is used for. Next, I will generate a list from my students about what they remember and together we will fill in any missing gaps.

Today’s task worksheet is pretty straightforward; students will be finding the missing parts of a triangle given Side-Side-Angle. My students usually never make the connection that there could be more than one possible triangle, so it is a good way to pique their interest once they realize that their answer is only half right. I have them work on #1 and #2 with their table and monitor the room and see if anyone picks up on the fact that angle B could have two possible values.

Question #3 is a review of another formula students learned in Geometry (Area = 1/2bcsinA). Not all of my students remember this one and may get stuck. If a student cannot figure out the area, I suggest that they use Area = 1/2bh and ask what value they are missing. Since they don’t know the height, they can add an auxiliary line to create a right triangle.

## Share

30 minutes

I always start by choosing a student who only found one measure for angle B to share their work. See if there is anyone in the class who disagrees or who wants to add anything to the student’s thinking. It is fun to build the dramatic tension by asking your students if they all agree with this solution and pausing for a long time. Then I will repeat the question a few times as needed. If I wait long enough, my students will know that something is amiss and will try to find our mistake. Once they realize that our work is not technically incorrect, they will be really intrigued and will be chomping at the bit to know what I am implying. My best students won't initially pick up on the fact that two triangles are possible, so it is powerful to have the whole class focused on one problem.

If a student realizes that there is another possible angle with the same sine value, I will have them share and explain their thinking to the class. If no one objects to the one possible triangle, I start by drawing the unit circle on the board. For this problem, we know sin(B) = 0.8, so mark 0.8 on the y-axis and ask students how we could find the angle that would produce this sine value. In the video below I walk through a set of questions you could ask to get them to see that there are two relevant angle measures that have a sine of 0.8.

Once it is out in the open that there is another possible angle for B, I will have them go back and work with their table group to find the other missing parts with the new angle measure. Also, I have them revisit question #2 and find the other possible angle. The triangle for question #2 will only have one possible angle measure since the angle is quadrant II is too large and will push the sum of the triangle’s measures greater than 180°. I don’t tell students this initially – let them investigate and come to that conclusion themselves.

This is one of my favorite applets to visually show why there could be two possible triangles when you are given SSA. Remind students about the geometric ramifications of the information you are given about a triangle and how you can know when a triangle is unique or not.

For Question #3, you can choose a student to present their work who has it written like Area = (1/2)(7)(3sin49°). The student will need to explain where the formula came from and how they found that the height of the triangle is 3sin49°. Then I ask students about what would happen if there were changes. What if the base was 15 instead of 7? What if the side length was 5 instead of 3? What is the angle measure was 82° instead of 49°. I will keep asking these questions until students see that the structure of the area expression will not change; then we can derive this area formula for a triangle. I will sometimes refer to this formula as the "SAS Area Formula" since you need two sides and the included angle to be able to plug into this formula. Giving it this name will be a reminder for students when it would appropriate to use.

## Summarize

5 minutes

To finish the lesson I want to summarize the three formulas we have worked with over the last two days (Law of Cosines, Law of Sines, and SAS Area Formula). As an Exit Ticket, you can have students hand these two questions  to you as you leave.

1. Draw a triangle with three pieces of information (side lengths and angle measures) that you could use the Law of Cosines with.

2. Draw a triangle with three pieces of information (side lengths and angle measures) that you could use the Law of Sines with.

3. Draw a triangle with three pieces of information (side lengths and angle measures) that you could use the SAS Area Formula with.

Evaluating the three formulas is not too difficult, so I really want the focus to be when to use the formulas. In my eyes that is the most important part - knowing that these tools are available and realizing when they apply to triangles we are working with).