Week 11

This week, we continued on in chapter 4, Trigonometric Functions.

On day 1, we looked at Graphs of Sine and Cosine Functions. 

An equation for Sine would look like:

y= a sin b (x-c) + d

a stands for= amplitude: amplitude is the max and min or how high it goes up

b stands for b in period: Period is 2π/b. A period is the length of one cycle. 

c stands for SlOP: Slop is slide opposite and if the number is negative. Move to the right and vice versa.

d stands for vertical shift: which is moving up or down on a graph.

On the second day, we continued on this and looked at how to find all x-intercepts of sine and cosine.

To find the x-intercepts for sine: Let bx+c = nπ
To find the x-intercepts for cosine: Let bx+c= π/2 + n

Below are the graphs of sine and cosine:

The Graphs are very similar but the only difference is where the x-intercepts are and where the graph starts.

On the next day we then looked at Other Trig Functions and we looked at Tangent.

The Tangent Graph is very different from sine and cosine as sine and cosine do not need asymptotes nor does it move upward for infiniti. The Tangent Graph moves upward and only at the origin does it change. Below is what the tangent graph would look like

y=tanθ

The graph makes a sudden twist at the origin and then goes on forever without touching the asymptotes and then more graphs are created to the right and left of each graph. 

the tangent equation is: y = a tan (bx+c) + d

To find the asymptotes of an equation you would use the equation:

π/2 + kπ

or -π/2b and π/2b

to find the period of the equation: π/b

To find the x-intercept you put (bx+c)=nπ

a is the amplitude and the amplitude of the equation is where the turn is created. For the equation above, the amplitude is 1 and so it turns at 1 and -1.

This is what was covered for the tangent graph.

The next trig function is cotangent. 

Cotangent is practically the same thing as tangent but there are a few or multiple differences. One difference is that cotangent goes in the opposite direction. 

For cotangent: the asymptotes of the graph is where the origin of each graph is. 

Below is a picture of cotangent

y= cotθ

What is suggested is that you graph the problem as if it was tangent and then convert it into cotangent

For example: y = 7 cot x

You would graph this as y = 7 tan x and then you convert it to cotangent the Graph is in the switched direction of the tangent graph drawn and the origin or where it meets the x-axis is also the asymptotes for a regular tan graph.

Cotangent is practically the same thing as tangent but switched.

Next is Secant. Secant is the opposite of cosine. 

The equation for secant is y = a sec (bx+c) + d

What makes secant different from all the other graphs is that it does not exactly connect just as tan and cotangent do not.

Below is a secant graph:

y= sec θ

In the blue is the original cosine graph and the walnut color is secant.

Secant practically is just parabolas over the turns of each one.

It is also recommended to sketch the cosine graph prior to actually drawing secant as that is the only way to find it. The asymptotes are where the lines touch the x-axis or the midpoint.

Whenever there is a vertical shift: draw a fake x-axis and draw asymptotes from where cosine touches it. Then you draw the secant graph.

The next trig function is very similar to secant but it is cosecant. It follows the sine graph instead and does exactly the same thing as secant and you would always draw a graph of sine first before putting the cosecant graph up. 

Below is a graph of the cosecant graph:

y = cscθ

It is exactly the same as secant but it is on a sine graph. You would continue to do this until you must stop graphing. This is how you solve a cosecant graph.

Those are the trig functions used and from these many other functions appear. 

This is what we did this week and HAPPY HALLOWEEN

SEE YOU NEXT WEEK!