Chapter 4.5 and 4.6

Okay so for 4.5 we learned about the basic since and cosine curves.The cosine and since graphs look like this:

So the x-coordinates on this graph represents the angle measure in radians on a unit circle, and the y-coordinates represents the points at which the angle equals the x-coordinate (if that makes sense). The cosine and sine functions are pretty much identical, except for the fact that the intercepts are different because they have 0s at different points on the graph. So as you can see from the graph, the sines and the cosines all have the same max and min, the same period length. The only difference is that the the y value of sine = 0 when x = 0, while the y value of cosine = 1 when x=0. This is because the sine represents the y value of the point on a unit circle, and

the cosine represents the x value of the point.

A period in the a graph is 2 pis long for a sine and cosine.

The equation for these graphs are in this format:

y = d+a sin(bx-c) or y = d+a cos(bx-c)

d moves the graph up and down

a vertically stretches and compresses a graph, meaning that it will increase or decrease the maximum and minimum, but leave the x intercepts alone.

b horizontally stretches and compresses a graph, meaning that it will stretch out the x intercepts or compress them together

c moves the graph left and right, but unlike d, c moves right when it is subtracting from x, and moves left when adding to x.

The next section, 4.6, was about the tangent function, cotangent, cosecant, and secant functions

The tangent function is the sine function over the cosine function, which looks like this:

This means that whenever the x makes the value of the cosine = 0, the answer for the tangent function will be undefined, meaning there will be an asymptote wherever cosine = 0. This also means that wherever sine = 0, the tangent value will also be 0, so that's where the graph will intercept the x axis.

Unlike cosine and sine, the period is only 1 pi.

The cotangent function is the cosine over sine, so for this when the sine value = 0, there is a asymptote and when the cosine value = 0, there is an intercept. They look like this:

Cosecant functions are parabolas that open up at the max and open down at the min of a sine function. Their asymptotes are when x= n(pi) and they look like this:

The equation for these are: csc x = 1/sinx

Finally, the last one is the secant function. This one is like the c

osecant, because it looks just like it except that it opens up and opens down at the max and min of a cosine function. Their asymptotes are when x = (pi) / 2 + n (pi)

The graph for this looks like this: