# LAW OF SINE - FINAL MATH PROJECT

0 (0 Likes / 0 Dislikes)

♪
Imagine for a moment a scene in a bank where a man threatens to shoot everyone dead
Imagine the criminal doesn't know about two of the hostages
In this sense, they are the only ones who can run outside and go for help
For now, let's call them A and B
For a moment let's forget about the criminal and just consider A and B
They might not know it
but WE know that between one another, is a distance of 5 meters
that they form an angle of 81° with the door
and that A is at 3.45 meters from the door
Now, if for a moment we say both of them run at the same speed
do you know which one has the best opportunity to run outside before the man can shoot them?
I guess the logic answer would be "does that even matter, just run for it!"
or "why don't you just use your cellphone and call the police?"
BUT
because this is math, let's unnecessarily complicate things
For this case, we'll use the LAW OF SINE
In trigonometry, the law of sine is an equation relating the lenghts of the sides of any shaped triangle to the sines of its angles
So, according to this law
in order to calculate lengths, we use
a/sinA = b/sinB = c/sinC
And to calculate angles, we use
sinA/a = sinB/b = sinC/c
Where the lower case refers to the lengths
and the upper case refers to the angles
So in order to answer the question and know who has a better opportunity of surviving
and to explain completely this law
Let's first search for the rest of the triangle's angles
So, if we have that sinB/b = sinC/c
then sinB/3.45m = sin81°/5m
Let's go step by step
sinB converts to sinB = (sin81°/5m)3.45m
which s equal to 0.68
And to have B alone, we convert it to
B = sine inverse of (0.68)
Which results in 42.96°, but to make it simpler, let's round it to 43°
And because I'm assuming that we all know by now that the addition of the internal angles of a triangle gives us a total of 180°
we can just add 81° plus 43°, equal to 124°
and then subtract the result from 180
this way we'll get A's angle, equal to 56°
From here, the only thing left is to calculate the distance between B and the door
which we do by comparing a/sinA = c/sinC
that will be the same as saying
a/sin56° = 5m/sin81°
This will convert to a = (5m/sin81°)sin56°
which gives us as result 4.19 meters
Se, we determine that A is the one who actually has a better chance of running out alive
*SHOTGUN FIRES*
but I guess we'll never know
if they made it
♪