## Unit 5 - Day 1

##### All Units
###### ​Learning Objectives​
• Discover the relationship between sides and their opposite angles in any triangle

• Identify the conditions needed to use the Law of Sine

• Solve for missing sides and angles using the Law of Sines

###### Experience First

We begin this unit by thinking once more about triangles, just as students did in Unit 4. This time, we look at oblique triangles and learn how the trig ratios can be applied in new ways to solve for sides and angles.

Students begin by finding the height of an oblique triangle, noticing that any triangle can be turned into two right triangles by drawing an altitude. This altitude is drawn for them in question 1. By noticing two equivalent ways of finding the height, students generalize that c sin B is equal to b sin C. As you are monitoring students, make sure students can articulate why these two expressions are equal. Also be looking for students who accidentally write b sin B=c sin C, as this is not a true relationship in the triangle.

In questions 3 and 4, students look at an obtuse triangle, first finding the height and then using the height to find the length of AB. They see that their generalized equation from question 2 still works. Surprisingly, we had many students say that the formula wouldn’t work because the right triangle is outside the triangle, and they used different angle measures to find the height (using the 56.3˚ supplementary angle instead of the 123.7˚ angle). They may require specific prompting to actually check the numbers and see that the relationship still holds. The debrief of this part of the lesson is a nice review of Unit 4 unit circle ideas, namely why the sin (56.3) would be equivalent to sin (123.7).

###### Formalize Later

When students generalize the relationship between the sides and angles of the triangle, they are likely not going to write the Law of Sines in its traditional format (as a proportion). Emphasize to students that sometimes they will see their discovered relationship written in a different way, but that both ways are completely equivalent. It’s important that students take ownership of their learning and aren’t shut down immediately with the “real” version of the Law of Sines. Since we want students to be critical thinkers, ask them why one might prefer one format of the equation over another. Students were quick to point out that using a proportion might make it easier to remember since the b and B are together and the c and C are together. This is a matter of preference and convention, not a matter of right and wrong.

The important ideas section will introduce the fact that these two ratios are also equal to a/sin A. Depending on time, you may wish to have students prove why this is true, by drawing the altitude from a different vertex and applying the transitive property.

Once you write the Law of Sines, ask students to consider what minimum information they would need to be given to be able to solve for a missing side or angle. Students should be able to articulate that they need at least one set of angle and opposite side to solve the proportion. This will lead to the generalization of why the Law of Sines works when given AAS, ASA, and SSA. In question 1 of the Check Your Understanding, students will see the validity of the ASA condition, even though it doesn’t initially appear that they have an angle and opposite side.

The experience portion of this lesson has students find the height of the triangle as a way of then finding the missing side using right triangle trig. The Law of Sines does both of these steps at once and thus eliminates the need of first splitting a triangle into two right triangles. If students continue to draw the height to solve for a missing side or angle, let them! We prefer students to use their own sense making before rushing to an algorithm. Over time we expect that students will transition to using just the Law of Sines once they see that it accomplishes the same thing in less steps.