BQ #6: Unit U

1) What is a continuity? What is a discontinuity?

A continuity is when a function proceeds without interruption, meaning that there are no discontinuities. Continuous functions are predictable, and one would be able to draw the function in its entirety without lifting their writing utensil. To extrapolate on my earlier point of "no discontinuities", I meant that the function contains no holes, breaks, or jumps - respectively, a missing point in the graph, a vertical asymptote which invalidates a specific x-value, or a random jump up or down in the function. A continuous function looks like this:

A discontinuity is when a disruption in the continuation of a function occurs. This may occur through either removable discontinuities - holes, or non-removable discontinuities - jumps, oscillating behavior, or unbounded behavior. Holes are when there is a missing point in the graph, jumps are when a graph drops/jumps, oscillating behavior is when there are wiggles, and unbounded behavior is when there is a vertical asymptote. The reason why we separate these into two families, is because removable discontinuities have limits, because their left/right limit values are the same, as we can see in the picture below. Therefore, the non-removable discontinuities must not have limits - due to different left/right limit values, no exact point, or unbounded behavior.

2) What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and a value?

A limit is an intended value of a function. It exists when a function is continuous, or when there is a jump discontinuity (a removable discontinuity). This is because as long as you can trace along the function from both sides and end up at a single, same point on the graph, there is an intended value. It doesn't exist when there is a non-removable discontinuity within the function - namely, jumps, oscillation, or unbounded behavior. This is because you can't end up at a point by tracing from left/right of the point, because of a sudden change in the y-value, no specific coordinate, or a restriction upon a value, respectively. The difference between a limit and a value is that a limit is intended, while the value is what is actually there. In the bottom example, the limit of the function is -2, but the actual value of the function is 2.

3) How do we evaluate limits numerically, graphically, and algebraically?

To evaluate a limit numerically, one would set up a table with values very near the point which you are trying to reach, and what they end up being after you plug them into the function. In the example below, you would reason that whatever is between the values which are very close to the intended point, is between the left/right close values, just as you would with any other numerical evaluation of a limit.

To evaluate a limit graphically, you would simply trace from left/right of the point, in order to find the limit. In the previous example, you would trace left/right of the x-value of 1, and end up ending up at 1 for your limit.

To evaluate a limit algebraically, you would utilize algebra skills in order to find a way in which you can plug in the number which x approaches to get something other than indeterminate form, or infinity of some sort, which are the only two restrictions for evaluating limits algebraically. The very first method in which you would try to evaluate a limit algebraically, would be plugging in the number which x approaches directly into the function. If that results in indeterminate form, or infinity, then you must move on to factoring, which would remedy the issue of indeterminate form. When you can't factor to prevent indeterminate form, you have to multiply by conjugates, usually for radicals, in order to get something other than indeterminate form. Lastly, if you have infinity of some sort, you would have to divide the entire function by the highest power variable in order to somehow fix the problem, and get an answer with substance - without the vagueness of  infinity.

BQ #4: Unit T Concept 3

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

Before we delve into the material, there are two important precursors to the logic that shall be put into explaining just why tangent graphs are typically uphill, while cotangent graphs are typically downhill.

Firstly, it's crucial to know that the ratios for tangent and cotangent, in terms of sine and cosine, are sin/cos, and cos/sin, respectively. From that, we would know that whenever cosine is zero, tangent shall be undefined, resulting in an asymptote, and the same goes for cotangent whenever sine is zero as well.

Secondly, we should know that the x-axis for these graphs is in terms of the unit circle - of π and all that. And, we know that in certain areas of the unit circle, tangent/cotangent are positive/negative. They're both positive in Quadrants I and III, and therefore, are negative in Quadrants II and IV. Since every π/2 on the x-axis is a quadrant, the areas where tangent/cotangent are positive/negative can be organized neatly into certain portions of the x-axis.

Basically, asymptotes divide these portions into repeating portions of either positive to negative (going from left to right), or negative to positive. The reason for which tangent and cotangent differ in where they are allowed to be positive/negative, is that their asymptotes are located in different places - where sine or cosine are zero.

In the above picture, there are asymptotes at π/2 and 3π/2, since that is where cosine is equal to zero. In the sections where tangent is allowed to continually exist for two sections of the graph, it goes from negative to positive, from left to right.

In the above picture, there are asymptotes at π and 2π, since that is where sine is equal to zero. In the sections where cotangent is allowed to continually exist for two sections of the graph, it goes from positive to negative, from left to right.

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.

Most simply, the graphs of sine and cosine relate to each of the others, in how the others' asymptotes would be where either sine or cosine are equal to zero. This is because sine is the denominator for cosecant and cotangent, and cosine, the denominator for secant and tangent.

Sine is equal to zero, when on the x-axis, it is at zero, and π, as illustrated by the graph below. This is so, because sine is y/r on the unit circle, and therefore, shall only be equal to zero when y is equal to zero. And, y, on the unit circle, is equal to zero, at those places - 0° (zero) and 180° (π). Therefore, there shall be asymptotes for cosecant and cotangent when sine is equal to zero. Asymptotes are important if you want to find out why cotangent goes down. It basically goes down, because the two areas where uninterrupted, consecutive points can continue from left to right, dictate that the left area shall always be positive, and that the right area shall be negative. This is because of the unit circle's ASTC thing. So, since cotangent has to go from positive to negative, it has to go downhill. It's the only way. The only way that I could possibly see sine relating to cosecant, is that they share the same positive/negative areas of the graph. In the areas where sine is positive, cosecant is positive, and vice-versa.

Cotangent is the orange line, and cosecant is the blue line.

Cosine is equal to zero, when on the x-axis, it is at π/2 and 3π/2, as illustrated by the graph below. This is so, because cosine is x/r on the unit circle, and therefore, shall only be equal to zero when x is equal to zero. And, x, on the unit circle, is equal to zero at those places - 90° (π/2) and 270° (3π/2). Therefore, there shall be asymptotes for secant and tangent when cosine is equal to zero. Asymptotes are important if you want to find out why tangent goes up. It basically goes up, because the two areas where interrupted, consecutive points can continue from left to right, dictate that the left area shall always be negative, and that the right area shall always be positive. This is because of the unit circle's ASTC thing. So, since tangent has to go from negative to positive, it has to go downhill. The only way that I could possibly see cosine relating to secant, is that they share the same positive/negative areas of the graph. In the areas where cosine is positive, secant is positive, and vice-versa.

Tangent is the orange line, and secant is the blue line.

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 do NOT have asymptotes, because their ratios on the unit circle are x (for cosine) and y (for sine), over r. And, r shall always be equal to 1, being the length of the radius of the circle. The logic behind that explaining how they don't have asymptotes, is that for there to be an asymptote, the denominator of a trigonometric function's ratio, has to be able to be equal to 0. Asymptotes only exist when there is a denominator that could possibly be equal to 0, therefore restricting a certain part of the graph. It is from that same requirement for there to be an asymptote, that we can reason that secant/cosecant/tangent/cotangent have asymptotes. Their ratios on the unit circle should be, respectively: r/x. r/y, y/x, and x/y. Since their denominators are not restricted towards being solely 1 as sine and cosine are, 0 is a possibility for the denominators. The end.

BQ #2: Unit T Intro

How do the trig graphs relate to the unit circle?
Trig graphs relate to the unit circle in how the unit circle can be utilized to find where a certain trig function is positive or negative on a graph. For example, sine is positive in the first and second quadrants - between 0° and 180°, or between 0 and π. If we utilize radians on our x-axis, and have some fraction of π as our markings, then we may isolate the area of the graph between 0 and π as the area in which sine would be positive. Finding these positive/negative areas works with every trigonometric function, though you would have to account for asymptotes as well for denominators that could possibly be equal to 0.

Period? 
Why is the period for sine and cosine 2π, whereas the period for tangent and cotangent is π?
The periods for sine and cosine are 2π because it takes a full rotation of the unit circle, which is 2π, for sine and cosine to fulfill the requirements to be periods. And, a period is a portion of any trig graph that is repeated over and over again without change. To illustrate this example, the values of sine, going through the unit circle's four different quadrants, should be +, +, -, -. Having anything less than those four repeating themselves would create inconsistency, so there you have it. On the other hand, the periods for tangent and cotangent are π because it takes half the rotation of the unit circle. π, for tangent and cotangent to continually repeat themselves consistently. The values of tangent/cotangent, going through the unit circle's four different quadrants, should be +, -, +, -. And having just two repeat themselves continually should be enough to constitute an acceptable pattern.

Amplitude?
How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?
Sine and cosine have amplitudes, while the other trig functions don't have amplitudes, because their ranges are restricted by how both of their trigonometric ratios have 'r' as the sole value in their denominators. And, on the unit circle, 'r' is equal to 1, being the length of the radius. The other trig functions don't have amplitudes because restrictions such as 'r' aren't in their ratios.

Reflection #1: Unit Q - Verifying Trig Functions

1. Verifying a trigonometric function means simplifying it and or replacing it with other trigonometric functions so that it may equal a set value. You would multiply/divide/add/subtract numbers or some other mathematical asset in order to advance with the equation and simplify it, until it arrives at the value you want it too equal. You may also replace trigonometric functions with other trigonometric functions, such as making tangent into sine divided by cosine. Or, you could make sine squared added with cosine squared, equal 1. One should really try to work with all of the trigonometric identities in order to help them arrive at what they want. And, it will always (at least for what we're doing) be equal. You're not trying to prove it wrong or right - rather, you're showing the work that it takes in order to prove it right. So, if you ever find adversity, don't pass off the equation as invalid.

2. I have found all of the tips and tricks equally helpful. The best tip/trick that I believe anyone could get, is that all of the equations are valid, so you can't give up and write it off as not equal. The most fundamental, and necessary (for about each and every problem you'll have to solve) is tip/trick, is that you have to utilize identities. Unless, of course, you get the problem tangent plus 1 equals tangent plus 2 minus 1, which would then necessitate the question of whether or not you are doing the right problem.

3. Basically, I'd look at what I could do: subtract/multiply by conjugate/replace, and do it. If I end up at a dead end, I'd try again, until I'd get it. And if I don't get it again, I shall try again.

P.S. Not giving up is the fundamental thought process for success in Unit Q.

SP #7: Unit Q Concept 2 - Finding All Trigonometric Functions Using Identities

This SP #7 was made in collaboration with Sergio Sanchez. Please visit the other awesome posts on their blog by going here.

Our given is:

One of the first and foremost steps that must be undertaken, is identifying which quadrant we are working with. Although, using the values of the trig functions and their accompanied signs would do just as well.

Next, we'd just go right into solving this with trigonometric identities. It's pretty self-explanatory had one taken this course. But, for the sake of something: first, I list out the identity, and then, substitute in the value(s) that I already have. I then solve it until there is one unknown trigonometric function is on one side, and a value is on the other.

As for Sergio's work, I suppose that I shall put forth some sort of explanation, in lieu of the original creator, who just gave me this picture. Basically, he plugged in values that were given (from the tangent trigonometric function), into the Pythagorean Theorem, in order to get the excluded side's value. Having gotten all of the sides, he just plugged them into the rest of the unknown trigonometric functions.

It is important that one remembers what all of the trigonometric functions mean - what sides they represent, and how they represent each other. As in, you have to remember that sine is opposite / hypotenuse, and also, 1 / secant. This is important because if you could fail pretty easily if you use the wrong functions/identities. It's also very important for one to remember to rationalize (which I did), and reduce (which I didn't do). Apparently, it's the most important thing you could do for yourself, for forgetting to do so, would result in loss of points on your tests or whatever.