BQ #4 Unit T- Concept 3

4. Why is a "normal" tangent graph uphill, but a "normal" cotangent graph is downhill? Use unit circle ratios to explain.

From Mrs. Kirch's SSS packet

tangent: Quadrant 1 is postive, 2 is negative, 3 is postive, and 4 is negative. The ratio for tangent is Tan(x)= y/x. So as you can see when cosine equal zero that means that there is an asymptote. There is an asymptote because when it is zero, it is undefined which means there is an asymptote where ever x equals zero. Tangent has asymptotes at pi/2 and 3pi/2 because that is where x equals zero. So as you can see in the picture above pi/2 is before quadrant 2 which is negative, so the graph will start at the bottom and work its way up because the quadrant before 3pi/2 is postive.

cotangent: The quadrant have the same signs as tangent but the ratio for cotangent is cot(x)=x/y. So as you can see, now it is when sine equals zero where their is an asymptote. There is an asymptote at zero and pi for cotangent because that is where y equals zero. So as you can see in the picture above, the quadrant after 0 is postive and the quadrant before pi is negative. So the graph will start on the top and then go downhill.

So tangent goes uphill and cotangent goes downhill because of the location of their asymptotes.

BQ #3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.

a. Tangent?

-Sine and cosine relate to cosine because of their signs on the unit cicrle. The ratio for tangent is tan(x)=Sin(x0/cos(x). So since the tangent ratio includes sine and cosine, their signs affect the tangent graph. So if sine is postive and cosine is negative, then the tangent group will be negative and go downhill. If sine and cosine is postive then tangent will be postive and going uphill.

b. Cotangent?

-Cotangent is just the reciporcal of tangent. Cotangent's ratio is cot(x)=cos(x)/sin(x). So cosine and sine's signs depend on what way the graph is going. So if sin is negative and cosine is postive then the graph will go downhill because it will make cotangent negative. We get these signs from the unit circle.  Plus it has diffferent asymptotes than tangent. 

c. Secant? 

-For secant, sine does not affect this graph at all. The only one that affects it is cosine. Cosine affects this graph because the ratio for secant is sec(x)=1/cos(x). So this means that cosine determine where the asymptotes go because cosine can equal 0. If cosine is 0 then it is undefined which means there are asymptotes. So cosine affectets secant because of the asymptotes.

d. Cosecant?

                            

                                     

-For cosecant, cosine does not affect this graph at all. The only one that affects it is sine. Sine affects this graph because the ratio for cosecant is csc(x)=1/sin(x). So this means that sine determines where the asymptotes go because sine can equal 0. If sine is 0 then it is undefined which means there are asymptotes. So sine affects cosecant because of the asymptotes. 

BQ #5: Unit T Concepts 1-3

Mrs. Kirch's SSS packet

5. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use Unit Circle ratios to explain. 

-Sine and Cosine are the only two trig functions that do NOT have asymptotes. The reason behind this is because asymptotes happen when you get undefined. The only way you can get undefined is when you divide by zero. According the the trig ratios, sine is y/r and cosine is x/r. So as you can see, they do not divide by zero because r is equal to one for the Unit Cicrle. Cosecant is r/y and cotangent is x/y so they always have the same asymptotes because sine can equal zero. Secant is r/x and tangent is y/x so they have the same asymptote because they both divide by cosine, and cosine can be zero so that means there is an asymptote present. 

BQ #2- Unit T Concept Intro

From Mrs. Kirch's awesome SSS packet 

1. How do trig graphs relate to the Unit Circle?

-Trig Graphs relate to the Unit Circle because they are basically the same thing. We just unwrap the unit cirlce and make it a line and it turns into a trig graph. It has the same pie values in the same four quadrants. Another reason why they relate is by the signs. The four quadrants stay the same as well. All the signs for each trig functions are the same, so we must remember ALL STUDENTS TAKE CALCULUS. As seen in the picture above you can see how the signs correlate with the unit circle.

a. Period? Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

-The period for sine and cosine is 2pie. The reason behind this is because the pattern for sine is postive postive negative negative. The pattern for cosine is postive negative negative postive. It takes 2pie for this pattern to repeat. So this makes it the period because these graphs go on forever and to repeat the period it takes 2pie. You can see a visual of the sine cosine trig graphs below.

The period for tangent and cotangent is pie. The pattern is postive negative postive negative. As you can see, the pattern repeats half way through the graph/ unit circle so that means that it is only pie and not 2pie because it repeats half way through the revelation. You can see a visial of the tangent trig graph below.

b. Amplitude? How does the fact that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?

-The fact that sine and cosine have amplitudes of one relate to the unit circle big time. On the Unit Circle we remember that sine and cosine could not be smaller than -1 and larger than 1. It relates to this because their amplitudes must be 1 as well. If it is greater than 1 or less than 1 it is considered undefined just like it was on the Unit Circle.

Reflection #1: Unit Q- Verifying Trig Functions

1. In this Unit, Unit Q, we learned how to verify a trig function. This actually means that we must prove that both sides are equal to each other because we know our answer already. So we must use our identities, which we all memorized to make the left side equal to the right side. WE NEVER TOUCH THE RIGHT SIDE. We are just verifying that is proven to be right. 

2. There are many tips and tricks that I have found helpful in this unit to help me understand it way more. One huge tip that I learned that if you can, change EVERYTHING into sine and cosine. By doing this it will make our lives easier because they will either cancel out or be equal to one. Another tip is to always work on the complicated side first that has a lot going on. By doing this first we are able to make it easier and get the hard side out of the way. A third tip is if you can, look for the greatest common factor. This will allow us to make our problems simplier and closer to our answer. The biggest tip I can give is to ALWAYS identify what you did in each step and make it neat. This will help you see where you went wrong if you did not get the answer right. It will also look nicer as well. 

3. The first step in my thought process is to see which identies I have in my problem. I always look to see if I have sine and cosine together, secant and tangent together, or cosecants and cotangents together because they pair up well with their idenities. Next I look for to see if anything cancel or if I have a greatest common factor so I can simplify it even more. If none of that works, the next step I do is look for a common denominator. Once I have done one or all of those steps, usually the identies work out the way I want it to and give me the same answer as the right side. Concepts 1 and 5 are tricky, but if you practice it will all make sense. Just memorize the identities. 

SP #7: Unit Q Concept 2

This SP#7 was made in collaboration with Kelsea Del Campo please visit their awesome blog by clicking here.

Using Identities:

Using SOCAHTOA: 

 In this student problem me and my partner made our own example from Unit Q Concept 2. In this first picture we showed how to find the values using Ratio Identities, Reciprocal Identities, and Pythagorean Identities. In the second picture we showed how to find the values usng SOCAHTOA. As you can see, we can find the values using both ways. In the first picture, using identities, you can see that we found all our values using different Identities and just substituting them into the problem. We used one identity to find another and so on. For the second picture, using SOCAHTOA, we just used the Unit Circle ratios from the previous unit. By doing these two pictures we have proven that identities can be associated with SOCAHTOA. 

I/D #3: Unit Q Concept 1- Using Fundamental Idenities to Simplify or Verify Expressions

1. An "identitiy" are proven facts and formulas that are ALWAYS true. The Pythagorean Theorem is a type of "identitiy" because we get proven facts and formulas that end up being true for any numbers we plug into it. With using the Pythagorean Theorem we use for this unit we use x, y, and r. We use these instead of a, b, and c because on a graph it when we graph it becomes x, y, and r.

As you see in the picture above all we had to do to get the Pythagorean Theorem equal to 1 is divide by r^2. Once we had r^2 it became equal to 1. From dividing by r^2 I noticed that we got x/r which is the ratio for cosine we used for the unit circle. I also noticed that we got y/r which is the ratio for sine we used for the unit circle. If we subsitiute those two for cos and sin we get cos^2x + sin^2x = 1. We can conclude that cos^2x + sin^2x = 1 beigins with the Pythagorean Theorem. It is referred to the Pythagorean Ideinity because we use the Pythagorean Theorem to get it.

As you can see in the picture above, I chose one of the "Magic 3" ordered pairs from the unit circle to show that it is true. The ordred pair used was the 60* angle one which is (1/2, rad3/2).

2.

The picture above shows that how to derive the identity with Secant and Tangent. First we had to divde everything by cos^2ø. Then that would cancel out the cos^2ø to make that be 1. Then we got sin^2ø/cos^2ø which we subtitute with tan^2ø. Lastly we have 1/cos^2ø and we substitute that with sec^2ø. Our final answer will look like tan^2ø + 1 = sec^2ø. 

The picture above shows that how to derive the idenity Cosecant and Cotangent. First we had to divdie everything by sin^2ø. Then that would cancel and make it be 1. After we got cos^2ø/sin^2ø which we substitute with cot^2ø. Lastly we have 1/sin^2ø which turns into csc^2ø. Our final answer will look like 1 + cot^2ø = csc^2ø.