# Law of Sines (& Area)

1) Take a look at the yellow right triangle on the left.
Write an equation that expresses the relationship among angle *B*, the triangle's height, and side *c*.

2) Rewrite this equation so that *height *is written in terms of side *c* and angle *B*.

3) Now consider the pink right triangle on the right. Write an equation that expresses the relationship among angle *C*, side *b*, and the triangle's height.

4) Rewrite this equation so that *height* is written in terms of side *b* and angle *C*.

5) Take your responses to questions (2) and (4) to write a new equation that expresses the relationship among *C*, *B*, *c*, and *b*. Write this equation so that *C* and *c* appear on one side of the equation and that *B* and *b* appear on the other.

6) Now drag the slider in the upper right hand corner. Now, given the fact that the length of segment *BC* would be denoted as *a *(it's just not drawn in the applet above), write an expression for the area of this original triangle in terms of *a*, *b*, and *C*.

7) Same question as in (6) above, but this time write the area of the triangle in terms of *a*, *c*, and *B*.

8) Suppose that dragging the first slider dropped a height from point *C* instead of point *A*. Answer questions (1) - (5) again, this time letting *c* serve as the base of this triangle (vs. side *a*). Notice anything interesting in your results?