Graphing Trig Identities

To determine the validity of an identity by graphing, one must graph each side of the given equation in y=. According to the Transitive property, if an identity both graphs would be equal. To determine if

(sinx) - (cosx) = (sinx) + (cosx) 

Is an identity, we plug each side into the calculator like this:

But we get this graph:

Which shows that the equations above is not an identity.
Now let's graphically prove the hypothetical identity

 cot(x)/sec(x)=csc(x)-sin(x)

First we plug in the left hand side, which graphs as red:

Then the right hand side, which graphs in blue- if we are unable to see anymore red, the equation is an identity. 

Though it is helpful, one cannot prove an identity using only graphing. Trig proofs are an algebraic process where one can prove a statement by setting right side exactly equal to left side. Graphing can present errors in inputting correct values as well as this: if an equation is slightly different and looks very similar on a graph, it is easy to mistake it for an identity. Ideally, one would use both graphing and algebra to determine the validity of the identity. 

Building a Roof

We are building a roof. Our blueprints mandate that the pitch of the roof should be 11/18, but that's all the information we have. We've also been given a beam to function as the run of the roof, which measures 22 feet. We draw a sample triangle and label it "Triangle 1".
One builder suggested we use the tangent of our sample triangle to find the angle of elevation. Surprisingly, he brought his calculator to the work site, so we utilized its inverse tangent function to find the angle: 31.4 degrees. Figure 1 shows his notes.

Since Triangle 1 and Triangle 2 are similar (with 3 congruent angles), we can use a ratio between the adjacent and opposite sides of Triangle 1 and set it equal to the ratio between the respective sides of Triangle 2.
Now we can make the height of Triangle 2 our variable x, and solve to get 6.72 feet. {Figure 2}

To find the rafter line, all that's left to do is use the Pythagorean theorem and plug in our rise and run values, as shown in Figure 3.

Now with each value laid out, we can cut and begin to construct this roof.

Err in the direction of Kindness

From a very young age, I was taught, as I assume most children are, skills in morality and conduct from my parents and older siblings. This was a necessary task, as I'm quite uncertain whether or not kids are born with natural empathy, though stepping inside a preschool classroom at 9:00 in the morning would have me thinking they're not. Regardless, all my compassion seems to have been taught, and I suppose I'm lucky that somewhere along the way it stuck.
The most useful thing I've been taught is compassion. It's a concept that my father especially has instilled in me for as long as I can remember, and it works along with empathy to force up this seemingly instinctual urge to aid those in need.
And I don't mean lending someone a pen or feeling sad about worldwide starvation. These are good things, undoubtedly, but they don't qualify as kindness, at least not to me, because you're supposed to do them. It's expected and the reason you do it is because you think you should.
Kindness started for me when I was forced to share or wash the dishes, and that's how it starts for everyone. You're told to do things for the sake of being kind but you don't really know what that means until you learn it for yourself.
My father has always been a kind and selfless person, and I admire that because it brings an illumination from within that makes him jolly and easygoing. He helps people for the sake of being kind and he always said, eyebrows raised and using his i-know-better voice, "when you do something kind, you should expect nothing in return". There is no reason for you to do something kind, but you do it anyway, and that is precisely what makes it kind. And the feeling you get when that light starts shining inside you is more than enough reason to be kind, but again you don't realize that until you know what kindness is.