BQ #4: Unit T Concept 3

Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?

   The "normal" tangent graph is uphill because of its asymptotes, which is located at pi/2 and 3pi/2 and so forth. The quadrants is positive, negative, positive, and negative. Tangent is uphill based on its asymptotes. Meanwhile, cotangent is downhill because its asymptotes is located at 0 and pi and so forth. In a way, it's asymptotes is "shifted" away from the "normal" tangent graph's asymptotes. Although the quadrants still apply the same for cotangent, it is a downhill because of where the asymptotes is located. Remember, tangent and cotangent have different asymptotes because of its ratio and the denominator in the ratio (sine or cosine will equal to 0).

BQ #3: Unit T Concepts 1-3

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

1. Tangent

   Tangent has a ratio of (sin/cos). When the tangent graph is labeled based on the quadrants of the Unit Circle,  the positive and negative sign of tangent is determined by the positive and negative sign of sine and cosine. If both sine and cosine is positive on the same quadrant, then tangent will be positive, ending up with an uphill on the graph. If sine is positive and cosine is negative on the same quadrant, then tangent will be negative, ending up with the downhill on the graph. If cosine is negative and sine is positive, then tangent will be negative, which will still have a downhill on the specific quadrant of the graph. If both sine and cosine is negative, then tangent will be positive, which will have an uphill on that specific quadrant of the graph. Tangent has asymptotes when cosine equals to 0. Cosine equals to 0 at pi/2 and 3pi/2, which is where the asymptotes are located and goes on continously at the graph.

2. Cotangent

   Cotangent has a ratio of (cos/sin). Similarly, the cotangent graph is based on positive and negative sign of cosine and sine. However, the difference between tangent and cotangent is that it will have a downhill shape in the beginning of its period rather than an uphill shape like tangent. The reason for its downhill shape is because of where the asymptotes are located. Cotangent has asymptotes when sine equals to 0. Sine equals to 0 at 0 and pi on the Unit Circle. Thus, this is where the asymptotes will be located for cotangent.

3. Secant

   Secant is the reciprocal of cosine and has a ratio of (1/cos). Because cosine is the denominator and can equal to 0, then secant will have asymptotes. Secant has asymptotes at pi/2 and 3pi/2 and so forth. If secant is positive, then the graph will go uphill. If secant is negative, then the graph will go downhill. In other words, if cosine is positive or negative on the graph, so will the secant.

4. Cosecant

   Cosecant is the reciprocal of sine and has a ratio of (1/sin). Because sine is the denominator and can equal to 0, then cosecant will have asymptotes. Cosecant has asymptotes at 0 and pi and so forth. If sine is positive, so will the shape of cosecant of the graph. In other words, if cosecant is positive or negative on the graph, sine will also have to be postive or negative respectively.

BQ #5: Unit T Concepts 1-3

Why do sines and cosine NOT have asymptotes, but the other four trig graphs do?

    Sine and cosine have a ratio of (y/r) and (x/r) respectively. The variable r equals to 1 on the Unit Circle, which is the reason why sine and cosine will never reach undefined since its denominator is 1. Cosecant and secant have a ratio of (r/y) and (r/x) respectively. Moreover, their ratios are reciprocals to sine and cosine. It is possible for cosecant and secant to have an undefined answer since the denominator does not equal to one. Tangent and cotangent have a ratio of (y/x) and (x/y) respectively. None of these two trigs have the variable "r" in their ratio. Thus, there is a likely chance that they will have an undefined answer. Remember, undefined equals asymptotes. All four trigs, except for sine and cosine, can divide by zero. This causes it to be undefined and have asymptotes on the graph.

Reference:

Mrs. Kirch's SSS Packet

BQ #2: Unit T Concept Intro

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

   The trig graphs relate to the Unit Circle because it uses the positive and negative values of the quadrants in order to define how a period for a certain trig may look like. Each trig has its own unique shape. Nevertheless, in a trig graph, the Unit Circle basically unfolds into a line. The points and  the radians from the Unit Circle is still very similar in a trig graph when reading from left to right. A sine trig graph shows an uphill from the two first quadrants on Unit Circle and then downhill for the remaining quadrants. Cosine shows an uphill, downhill, and then uphill again because of the positive, negative, negative, and positive value on the quadrants. Tangent/cotangent both have an uphill and downhill. The trig graph for tangent/cotangent have a shorter period because the positive and negative on the first two quadrants is similarly repeated again in the third and fourth quadrant.

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

   On a Unit Circle, sine begins with a positive, positive, negative, and negative value on the quadrants accordingly. Cosine begins with a positive, negative, negative, and positive value on the quadrants accordingly. The period for sine and cosine is 2pi because it takes four quadrants to repeat the same pattern over and over again. Meanwhile, tangent/cotangent has a period of pi because it already has a positive and a negative value on the first two quadrants on the Unit Circle. Thus, it only takes pi to repeat the pattern over and over again.

3. 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 of one because in the Unit Circle, sine and cosine cannot be greater than 1 or less than 1. Otherwise, it will be considered as "error" or "undefined". Since sine and cosine has a ratio of y/r and x/r respectively (and r = 1), sine and cosine can only fit under the rule of greater or less than 1 and
-1.

Reference
Mrs. Kirch's SSS Packet

Reflection #1: Unit Q: Verifying Trig Identities

1. What does it  mean to verify a trig function?

    To verify a trig function means to be able to get a specific answer according to what's given within the problem. Verifying a trig function takes quite an amount of time since your goal is to try different steps and plug in any Ratios, Reciprocals, or Pythagorean Identities into the problem. Remember, verifying means that your answer or final result needs to match to what is given to you. Verifying can go along with the same steps as simplifying.

2. What tips and tricks have you found helpful?

   The most helpful tip there is to try to change your trigs in terms of sine and cosine. Another helpful tip is to memorize all of the Ratios, Reciprocals, and Pythagorean Identities. At times, it is helpful if you separate the monomial fractions or combine the binomial fractions. When in doubt, you can always take the greatest common factor or least common denominator so in the end, your denominator can cancel.

3. Thought process and steps when verifying a trig function

   Personally, when I verify a trig function, I tend to change the trigs into sine and cosine. If this step does not work and my denominator is a binomial, then I would do the conjugate to the denominator and numerator. My other option when verifying a trig function is to separate the fractions (if any) and change it in terms of sine and cosine. It is also very useful to take the greatest common factor because there might be a chance when that GCF can cancel.