# Week 17 – precalculus 11

The Sine Law

For the longest time in math we were told we had to do multiple steps to find the side length or angle of a point of a triangle. If it was not a right angle we would have to cut it in half and find the values and sometimes double them to get the whole triangle. The Sine Law changes all of this. As long as we have the side length and its angle as well as another angle we could find the second angles side length. We can also flip it around to find a missing angle. The Sine Law looks like so $\frac {a}{sin A} = \frac {b}{sin B} = \frac {c}{sin C}$ We can use any two of these values and make them equal each other to find the variable. The lower case letters stand for the side length, opposite to its angle, which is shown on the bottom. Now this is the format we use to find side lengths, but to find angles we use the recipricals or flip it. Giving us $\frac {sin A}{a} = \frac {sin B}{b} = \frac {sin C}{c}$. The reason we did not use this equation before is because we did not know how to calculate past 90 degrees, but because of the fact that we now know that triangles are on a circular plane of four quadrants, we now know how it works. I will now demonstrate how the sine law works with two simple equations.

We want to find side b, now we know that A will have no involvment so we will use just the b and c parts of the equation. We will now fill in what we know. We know that side c is across from angle c.

$\frac {b}{sin 30} = \frac {5}{sin 80}$

Then we will isolate b, giving us

$\frac {b}{sin 30} = 5.077133059$

Then we will multiply by sin 30

b= 2.5 cm the side is 2.5 cm long

Now we can find a missing angle in a similar way. For this next one we will have to do multiple steps.

Lets say we are trying to find angle A. The problem arises when we realize we don’t have the side a’s length. This does not mean we cannot answer the question. We just need to get creative. We will find angle B first.

So we will us $\frac {sin B}{b} = \frac {sin C}{c}$

We will insert our values

$\frac {sin B}{6} = \frac {sin 70}{9}$

We will be left with sin b = .626461747 This is because we divide the left side and then multiply by 6. We now need to get rid of the sin so we will use the inverse on the other side.

Giving us : B =39 degrees

We know that all the angles must add up to 180 degrees so we will take away 39 and 70 from 180. Giving us 71 Degrees. This is the angle A. We could even find side a using this method. All we would have to do know is use the equation once more. We will use the a and b portions of the equation

$\frac {a}{sin 71} = \frac {6}{sin 39}$

Then we will end up with 9 cm, after we divide and multiply to isolate a.

The sine law makes finding angles and sides much easier. Even if we do not have all the information we can use what we have to get it, to solve the problem.