BQ #3: Unit T Concept 1-3

How do graphs of sine and cosine relate to the others? Emphasize asymptotes.
For the rest of our lives we are going to refer to sine and cosine as being our "favorite cousins." Why, you might ask? Because they relate to our other trig functions through their ratios! By knowing this we already know that they will have some grand effect on the other four trigs.

Tangent
We know that Tangent's ratio is... y/x! Which, when referring to the unit circle values (sin = y/1 & cos = x/1), is equivalent to Sin/Cos. If we imagine the unit circle being unfolded to lay as the graph, we can picture the four sections that represent four quadrants; at the same time, we would place both sine ans cosine in the graph segment as well.  The area between zero and pi/2 would be known as Quadrant one. if we focus specifically on where sine and cosine are, we see that both are abooooove the x-axis! Therefore, because of tangents ratio, we know that tangent is also positive n the first quadrant. Curious about where the we get the placements of the asymptotes? Easy! Where there is an undefined value, there is an asymptote and we'd only get undefined by dividing by ZERO! Ergo, asymptotes will be wherever cosine is at zero.

Cotangent
Cotangent is very similar to tangent because they are what? Reciprocals! This means that the ratio for cotangent would be x/y because sine is the value of y and cosine the value of x. Again, we know that cosine, sine, and cotangent will intersect on the graph because of that one ratio. Now, we have to figure out the asymptotes. Because y is the denominator, we noe have to figure out where sine is at zero because it would create an undefined value therefore an asymptote! The only places where y equals zero is at (1,0) and (-1,0). Here, the radians are at zero and pi.

Secant
Secant is a little more different than tangent and cotangent because its period does not end at pi, but at two-pi which makes a complete revolution. Secants ratio is 1/cos. Although it takes longer, we can tell the positive or negative sign through the same method. If cosine is positive then of course secant will be positive as well! Our asymptotes for secant will always be the same as the asymptotes for tangent because they both have cosine as their denominator! However, because secant only has cosine in its ratio, instead of having a normal curve for tangent, their graph becomes a parabola facing the direction of positive or negative, depending on the quadrant.

Cosecant
With cosecant, we are going to see very similar things as we did with secant. The only difference is the value in its ratio. the ration for Cosecant is 1/sine. This means that the asymptotes for cosecant and cotangent will be equivalent because they too have the same denominator. When drawing a cosecant or secant graph, we always start with drawing its reciprocal FIRST because it is the general foundation of this graph, After drawing the reciprocal, we draw the asymptotes. For cosecant our asymptotes will be wherever sine is zero! From there, we draw the parabolas starting at the maximums and minimums of each hill on the reciprocal.