BQ 3 Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others?

a. Tangent. 

These graphs are different because we know that sin and cos are y/r and x/r so this is why neither of these graphs have asymptotes, r=1. Tangent on the other hand is y/x therefore wherever x or Cos is equal to zero, there is an asymptote.

b. Cotangent 

Similar to tangent, cot is just x/y so wherever y or sin is equal to zero there is an asymptote, this is because the graph is undefined. 

c. Secant 

Because sec is r/x it will have asymptotes at the same places where tan had them but the two graphs look different because tan is only pi for a period and sec is relates to cos graph which is 2pi for a period. 

d. Cosecant 

Because csc is r/y it will have asymptotes at the same place where tan had them, and once again they are different because of the difference in the period length. 

BQ 5 Unit T Concepts 1-3

An asymptote is when the value of the graph is undefined, we know that something is undefined when it is divided by zero. Now let's take a look at our ratio identities tan is equal to y/x, cot equal to x/y, secant is r/x and cosecant is r/y, if we look at al of these we notice that these mean the numerator are divided by a number ;x, y these could be any number including 0. Meaning that any one of these could have an asymptote. If we look at cos and sin their ratio identities are y/r and x/r r will always equal one and therefore these will never be divided by 0, they won't have an asymptote.

Tan= y/x                Sin= y/r
Cot= x/y                 Cos= x/r
Sec= r/x
Csc= r/y

BQ #2: Unit T Concept Intro

a) Why are the periods for sin and cos 2pi and the period for tan only pi?

If we look at all the signs that we have when looking at a sin graph, we can see that the order of this is; +,+,-,- this pattern does not repeat, therefore we need to have another cycle and then it will repeat, each of these cycles is equal to pi, therefore this graph's period will be 2pi

As for the cosine graph, the cycle is +,-,-,+ so we can see once again that the signs don't repeat themselves yet, therefore we had to once again add another cycle and make this period be 2pi.

When we look at the tan/cot graph however, we see that the cycle is +,-,+,- in this case the signs do repeat themselves so we have our pattern in only one cycle this makes a period of only pi.

b) If we think back to our ratio identities we know that sin is really y/r and cos is x/r we also know that r is always equal to 1 and then we can think about all our other ratios, tan y/x csc r/y ect r is no longer on the bottom, this means that the function isn't restricted to just one. Sin and cos are restricted because they will always be over r and r will always be equal to one, this is also why they have amplitudes of one.

Reflection #1 Unit Q: Verifying Trig Identities

1. To verify an identity means to try and get the left side of a problem equal to the right, to do this we usually have to simplify of use our ratios in order to make things cancel and reduce. Although the problem look like two different trig function when you begin, we are testing to verify that they can be reduced to equal the same thing.

2. Some tips that I have found helpful are to always think of things in terms of sin and cos, things may not seem similar until we convert them to these. I  have also learned to keep in mind your pythagorean identities while working a problem out, so that we are sure not to change the ones we need, or so that we have a trig function we are trying to get to.

3. My first thought when looking at a trig problem is to see if I have any parts for my identities (are things squared, to i have one or two of the ratios needed?) next I check to see if the problem would be easier if I changed it into cos and sin. If these don't work I look to check if anything can be factored out or foiled ect. My final thought is to see if I square the problem would I have a solution or would the problem become easier when squared.