BQ#4: Unit T Concepts 1-3

Why is a "normal" tangent graph uphill, but a "normal" tangent graph downhill? Use unit circle ratios to explain.
 The reason as to why these graphs are completely different is because of their asymptotes and their location. The graphs themselves are based on their Unit Circle ratios, and since the graphs can't touch the asymptotes because we have to remember that the asymptotes themselves are based on the Unit Circle.

Both tangent and cotangent have their asymptotes based on their Unit Circle ratios. Remember the ratio for tangent is y/x and in order to show an asymptote x has to be divided by 0 so it can be undefined. If we look at the graph, we see that 90 degrees and 270 degrees has the ratio to be undefined. 90 and 270 degrees in radians are pi/2 and 3pi/2, and those are the asymptotes. The same thing applies for cotangent except that the ratios are switched, it's x/y. So in order for cotangent to have asymptotes, y must be divided by 0 and 180 and 360 degrees have those points. The asymptotes for cotangent, in radians, is pi and 2pi.

Tangent:

If you observe a tangent graph already graphed we can see that 2pi and 3pi/2 are the asymptotes and the graphs can't touch them. Also, based on the Unit Circle, the four quadrants are present and if they are positive and negative. Both quadrants 1 and 3 for tangent are positive so it's above the x-axis. Quadrants 2 and 4 are negative so the graph is below the x-axis.

Cotangent:

The asymptotes for cotangent are pi and 2pi and just like Tangent, remember the graphs CAN"T touch them. The Unit Circle and it's four quadrants are just like tangent, 1 and 3 is positive and 2 and 4 are negative. Except that the graphs of cotangent will not be the same as Tangent because of the asymptotes make the graph go downhill to follow the positive- negative rule of the quadrants.

BQ# 3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each trig graph?
Sine will be in pink with a dotted black line.
Cosine will be in Pink.
Tangent will be in Blue.
Cotangent will be Yellow.
Cosecant will be in Purple.
Secant will be in gray.

Tangent:

To observe where the tangent graph will be we must first know about it's ratio. Tangent's ratio is y/x or sin/cos, cosine will determine where tangent will be. Where sine and cosine are both positive or negative then tangent will just be positive. If one of them is negative then tangent is negative. Tangent's asymptotes is determined by cosine, if cosine is 0 on the x-axis then that is one of tangent's asymptote. 

Cotangent:

Same thing goes for cotangent except that it's sine that needs to be 0. If both sine and cosine are positive or negative then cotangent will be positive. If one of them is negative then cotangent is negative. The reason why cotangent is downhill is because of the asymptotes and where they are found. Since it has to be positive in quadrant 1, the cotangent graph has to be above the x-axis. Same thing for quadrant 2 but it's negative so it's below the x-axis.

Cosecant:

Where sine is 0, the asymptotes for secant will be those points. If sine is positive in two quadrants then cosecant will be positive, if it's negative, like in quadrants 3 and 4, then it's negative. Cosecant relates to the graphs by it's shape, the shape is between two asymptotes. So the asymptotes basically determine the shape and the asymptotes is where sine is 0.

Secant:

Where cosine is 0 the asymptotes of secant can be found. The way the secant graph looks like, positive or negative, is the same as the cosine graph. The shape of this is also determined by the location of the asymptotes and the cosine graph. As you can see the graphs are quite set up, (sorry for the color mixup) v.v

BQ# 5: Unit T Concepts 1-3

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain. 
                 
            Sine and cosine graphs don't have asymptotes because asymptotes occur when the denominator of the ratios is 0 (undefined). The other trig graphs can be divided by 0 and it can be undefined while sine and cosine can't be divided by 0. They can only be divided by 1 because 1 is their restriction on the Unit Circle and on the Unit Circle itself it goes from 1 to -1 on both axis. They both can't be divided by 0 because it's not undefined but no solution. The other four graphs especially tangent and cotangent have no restrictions and if their ratios equal undefined then it's their asymptote.

BQ#2: Unit T Concept Intro

A)   When you think about sine and cosine, we have to remember their periods are 2pi and it's like that due to their similarity to the Unit Circle.



In the Unit circle Sine is positive in the first and second quadrant and negative in the third and fourth quadrant. If we start from zero and do one complete rotation to get it to be positive again, then that is a 360 degree or 2pi rotation. So when we stretch the Unit Circle into a line, the cycles will be stretched out too because the first two quadrants will be above the x-axis and the last two will be below the x-axis. 

 

 

 





This also applies for cosine since quadrant I and IV are positive and quadrant II and III are negative. We need to start at 0 degrees and as we move around in the Unit Circle well reach the positive when we get to 360 degrees. When it reaches 360 degrees the it made one complete rotation. If we again stretch out the Unit Circle to a straight line for cosine then we will start above the x-axis since quad. I is positive and for quads II and III it will be below the x-axis. As we approach quad IV to make a cyclical then it go back up above the x-axis.

 

Tangent is different because it's only half the trip to get a cyclical. Quadrants I is positive and quadrant II is a negative so we already have our cyclical. On the Unit Circle, it will start at 0 degrees and when we reach a positive again it will be at 180 degrees and the radian value is pi.



Reflection #1 - Unit Q: Verifying Trig Identities

1. What does it actually mean to verify a trig identity?
1. To verify an identity to me is to break down a complex problem to its simplest form, and to make sure that form is equivalent to the one you already were given, so it matches. These types of problems are very helpful as a check method to do, because if you know or think you understand the material, the more you verify these problems, and get them right, you'll notice patterns start to develope.

2. What tips and tricks have you found helpful?

The tricks and or tips i found the most helpful was listening to Mrs. Kirch. Memorize the Circle Chart and all the identities. It really helps and its crucial not only to do well on the test, but its more time efficient, plus you look like a total genius if you do ^.^

3. Explain your thought process and steps you take in verifying a trig identity.

My thought process actually varies from problem to problem because its like starting all over again so i have to go through a check list, such as, can i put it in sin or cosine, can i do the conjugate, etc etc. so the questions build up, and if your not careful, you can end up confused, the trick is for every question i ask myself, i try to answer it, or prove it whether its right or wrong. Trail and error is a good way to describe my actual process, but it never hurts to know your identities and your pi chart, that is, in my opinion, the most crucial thing out there.

SP #7: Unit Q COncept 2

In this SP I will show how to write and solve  a problem using two methods: The Identities, ( Ratio, Rational, Pythagorean) and the famous SOHCAHTOA.

SOHCAHTOA

In the image above you can see the work for finding the 6 trig functions using the method of SOH CAH TOA. To find the 6 trig functions we use a certain method of SOH which the OH in SOH is represented opposite over hypotenuse, and we substitute it in for the values/ numbers. TO find the CAH or TOH we use the same logical reasoning. As a result we were able to find the reciprocal to find the other 3 trig functions, which where CSC, SEC. COT.

IDENTITIES

In the identity method to find the 6 trig functions, we had to use the ratio, rational, and Pythagorean identities to solve. First we started of with tangent, which was SINE over COSINE, and basic substiitution of numbers,from there we were able to plug in COT, which equaled to 1/ tangent. The we substituted and we got our answer. Then we used the Pythagorean identity of 1 + tan ^2 theta = Sec ^2 of theta, then we substituted in for tan ^2 theta and then from there we combined like terms to get our answer. To find CSC, we used another Pythagorean identity. Which was 1 + COT ^2 theta = CSC ^2 theta and simplified the same way as finding for SEC.Finally we used the reciprocal identity of CSC and SEC to find SINE and COSINE, from there we basically simplified.

I/D #3: Unit Q: Pythagorean Identities

Inquiry Activity Summary

1. Where does sin ^2x + cos^2x = 1 come from?

Back in Unit N, we learned about the Unit Circle. In the Unit Circle we learned that the ratio for cosine is (x/r) and the ratio of sine is(y/r). Now, we all know the Pythagorean Theorem which is a^2 + b^2 = c^2,  but with the Unit Circle it would be x^2 + y^2 = r^2, but in the Unit Circle R always equals 1 and to make this true and correct, we would divide by r^2 on both sides, leaving us with (x/r)^2 + (y/r)^2 = 1.  So we can safely say (x/r) and (y/r)  are cosine and sine.  They're squared and because they are squared, we have Pythagorean Identity. This means it cannot be "powered up"or "powered down", also because it is a Pythagorean Identity it must be squared and no power greater or less. Also because the Pythagorean Theorem is a proven fact and the formula is always true, it is called an identity. We can also prove this by demonstrating one of the "Magic 3" ordered pair from the Unit Circle (30*, 45*, 60*). We'll use 6* now theta of 60* is (1/2, radical 3/2) with 1/2 being x and radical 3/2 being Y. To prove our derivation of x^2 + y^2 = r^2 we're going to use cos^2 +sin^2 = 1, so (1/2)^2 + (radical 3/2) ^2 =1.

To derive the remaining two Pythogorean Identities we have to divide the whole thing by cos^2x to find the tangent derivation which will lead us to tan^2x + 1 = sec^2x

Now all we divide the original by sin^2x to find the cotangent derivation leading us to 1 + cot^2x = csc^2x.

Now all this information was found from page 1 of our Unit Q SSS packet, which is information we should Memorize!

Inquiry Activity Reflection

1. The connections that Iv'e seen between Units N, O, P, and Q are:

1. First the role of sine and cosine in Unit Q from the Unit Circle and its properties which can be seen in the reciprocal identities which are similar to the inverse of these trig. functions from Unit O and P. Also it helps to see how all the units are coming along together. 
2. In Unit P Concept 3 when we are using the distance formula for Law of Cosines with SSS or SAS, one of the formulas is a^2 = c^2(sin^2A + cos^2A) - 2bcCosA + b^2 and we know that cos^2A + sin^2A = 1. Helps everything make sense :D 

2. If I had to describe trigonometry in THREE words, they would be: 

challenging, tiring, and confusing.