Proving Trig Identities

Proving Trig Identities

An identity, is an equation that is true for all "X" values.  Calculators can help us prove when an equation IS NOT an identity. However calculators cannot prove if an equation IS an identity. This is because a graphing calculator cannot display all values of "X" at one time. The display window in restricted to a finite number of values. This is problematic because if there is even one point where the two lines aren't the same outside the view able window, it wouldn't be an identity. To prove my above-stated point, I will prove that a trig equation isn't an identity, and show that a trig identity can't be proven with a calculator, but algebraically.

In sum:
a. Find an equation that is not an identity.  Use your calculator to demonstrate that it's not an identity. Be creative. Take screen shots, pictures, images, etc.b. Find an identity.  Use your calculator to show what an identity looks like on the screen. Be creative. Take screen shots, pictures, images,etc.c. Why can you not prove an identity using the graphing or tables tools? Explain.

A. I think we can all agree that tanx/secx=csx/tanx isn't true. It would be an identity if all values of x were the same for both graphs were the same(the lines would be the same). 

Obviously this isn't an identity and this can be proved by the calculator.

B. Let's take the equation (cot^2x)7 = (1 + cosx^2)7 



















so while this looks like an identity, we cannot prove it with just this image. What if there where an inconsistency outside of the view-able frame? Then it wouldn't be an identity. So, we have to prove it algebraically. This is done as such:

Trigonometry and houses

Without Trig, your house would be lopsided.

Architects and contractors use trig to find the "pitch" of a roof while road constructors use the term "grade". To understand "pitch" and "slope" we must understand the trigonometric function tangent. When talking about the pitch of a roof, a fraction is used to describe it. For instance 11/18. For every 11 inches rise, there are 18 inches of run. Rise/Run or Opposite/Adjacent is the notation for tangent as well!


So if the full length of the "run" is 22 feet, lets calculate the full length of the rise, and then the angle of elevation.
11/18 = x/22, x = 36.
So if the angle of elevation is Rise/Run, then 36/22 is the slope! The tangent function works such, that if you plug in the angle of elevation, that we would get the slope. The inverse of the tangent function however will provide the angle if you provide the slope. So to find the angle of elevation, we plug our slope into the inverse tangent function.
tan^-1(36/22) = 58.6 degrees
reference:
http://jwilson.coe.uga.edu/EMAT6680Su09/King/Roofing/Application%20of%20%20Mathematics%20in%20Construction.htm

I was moved by Mr. Hoffman talking about all the interesting women he had never met because of their appearance. I don't think this only applies to people that one would be sexually attracted to, but to all. Not every person will be interesting to you, but to give everyone a chance to make an impression on you is important. We ask ourselves why we aren't more open, kind people. Why we don't judge someone by their appearance. George Syracuse answers this. We are self interested entities. And as we grow older we regret the missed opportunities to learn a person's story and bond with them. This "brainwash" that Hoffman talks about is all too apparent in present society especially in high school. It's frustrating. To see a judgement made about someone before even interacting with them. I think it is this frustration that should drive us to be kinder people.