Week 17 in Precalc 11

This week in math we learned about the Sine Law. The Sine Law includes an equation which you can use to solve triangles, as long as you have three clues: either 2 side lengths and an angle or two angles and a side length. You also need a matching pair of a side length and angle. Remember: the matching side of an angle is the side across from it, same for sides, the matching angle is the one across from it.

The equation we use has two different forms, one for when you’re trying to find a side length and another for when you’re trying to find an angle so that it’s easier to solve since the missing variable will be a numerator. For side-lengths: $\frac{a}{Sin A}$ = $\frac{b}{Sin B}$ = $\frac{c}{Sin C}$. For angles: $\frac{Sin A}{a}$ = $\frac{Sin B}{b}$ = $\frac{Sin C}{c}$. The letters can be exchanged if the question you are answering using different letters.

To start solving, put all the information the question gives you into the proper equation for what you’re trying to find. One fraction should have no variables, another should only have 1 variable, and the third fraction should have both missing variables. As long as you are solving to find the variable that is in the fraction with only 1 missing variable, you can cross out the third fraction with both missing variables and ignore it. This way you can use algebra to rearrange the new equation with only one equal sign and two fractions and then solve using a calculator to find the one missing variable.

If the variable you are trying to find is in the equation where both variables are missing, then you have to ignore that fraction first and solve for the other variable which should be an angle (refer to the previous paragraph). Once you have the amount for this angle, you can do 180=(angle given in the question)+(angle found) and solve to find the third angle. Now you have enough information so that there is only one variable in the fraction we are trying to solve, so you can solve for the variable. Now there’s your answer. There will be other variations of these questions where you’re solving for something different, be aware of which equation you are using and what you are trying to find at all times. Apply this knowledge to different questions, and you will find the Sine Law easy. I’ve included an example below to help you visually picture this process.