Real World Problems

You are lying next to the pool on your favorite long chair. You want to figure out what the angle is of the bottom of your long chair to the ground. You take out your protractor and do a quick measurement. The angle is 75 degrees. Now its not over. You decide to plug it into a sum and difference formula for sine.

Another...

You are designing a chair for your sick teacher because you love your teacher that much. While drawing up the blue prints you come across a problem, the angles of the chair are not adding up correctly. You need an exact measurement. Guess what that means? Double angle formula! The first angle you need to figure out is 75 degree angle. You must find the cosine of this angle. 

First, find two easily solvable angles that add up to 75.
Then plug them into the double angle formula.
Substitute the angles with the points on a circle
Now multiply together and simplify!

And Another....

You stumble upon an angle of the chair that is too small to have two easily solvable angles add up to it. You can't use the double angle formula anymore.
But you can use the half angle formula that your favorite math teacher taught you! Your protractor tells you that the angle measures
out to be 15 degrees. You should find the tangent of this degree.

So first ask your self, what is 15 times two? Thankfully it is an angle that can easily be solved. Now plug it into the half angle formula for tangent and plug in the points on the circle for 30 degrees. Now simply find a common denominator and copy change flip. After that, you have the answer.