BQ #3: Unit T Concepts 1-3

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

a. Tangent? The graph of tangent has its asymptotes based on cosine. To understand how this works, one must know that asymptotes are found in a graph when the trigonometric function is undefined. The trig ratio for tangent is sine/cosine, and we know that it is undefined when the denominator is equal to zero. So, if cos(x)=0, tangent becomes undefined, and thus it has asymptotes. Cosine equals zero at pi/2 and 3pi/2, so we know that two of our asymptotes lie there. Also, the positive and negative values of tangent are influenced by sine and cosine. For example, in quadrant two, sine is positive while cosine is negative. When we divide sine by cosine, since it is tangent's ratio, we get a negative value. Thus, we find that tangent is negative in quadrant two, which is true. This method applies for each quadrant around the unit circle.

b. Cotangent? The graph of cotangent has its asymptotes based on sine. The trig ratio for cotangent is cosine/sine, and we know that it is undefined when the denominator is equal to zero (thus giving us asymptotes). So, if sine(x)=0, cotangent becomes undefined, and then it has asymptotes. Sine equals zero at 0 and pi, so we know that two of our asymptotes lie there. Also, the positive and negative values of cotangent are influenced by sine and cosine because of its ratio. For example, in quadrant three, both sine and cosine are positive. When we divide cosine by sine, since it is cotangent's ratio, we get a positive value (a negative divided by a negative equals a positive). Thus, we find that cotangent is positive in quadrant three, which turns out to be true. This method applies for each quadrant around the unit circle.

c. Secant? The graph of secant has its asymptotes based on cosine, since it is its reciprocal. The trig ratio for secant is 1/cosine, and we know that it becomes undefined when the denominator is equal to zero (thus giving us asymptotes). So, if cosine(x)=0, secant becomes undefined, and then it has its asymptotes. Cosine equals zero at pi/2 and 3pi/2, so we know that two of our asymptotes lie there. Also, the positive and negative values of secant are also influenced by cosine, since they share the same ones.

d. Cosecant? The graph of cosecant has its asymptotes based on sine, since it is its reciprocal. The trig ratio for cosecant is 1/sine, and we know that it becomes undefined when the denominator is equal to zero (thus giving us asymptotes). So, if sine(x)=0, cosecant becomes undefined, and there we have our asymptotes. Sine equals zero at 0 and pi, so we know that we have two of our asymptotes there. Also, the positive and negative values of secant are also influenced by sine, since they share the same ones.

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.
The tangent graph is uphill while the cotangent graph is downhill due to their asymptotes, and this is because of their different trig ratios. Tangent's is y/x, while cotangent's is x/y. As we have learned in the previous concepts, asymptotes exist where the trig function of the graph is undefined, or when its denominator is equal to zero. Therefore, tangent would be undefined whenever x=0 (pi/2 and 3pi/2), while cotangent would be undefined whenever y=0 (0 and pi).

Even though the signs may be the same for the two graphs--positive-positive-negative-negative--tangent and cotangent graphs are different because their asymptotes are in different places. Thus, their whole graphs are "shifted" and this results in tangent going uphill and cotangent going downhill according to the signs.

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.
Asymptotes appear on graphs where the values are undefined, or when a ratio is divided by zero. The reason that the sine and cosine graphs do not have any asymptotes is because, according to their unit circle ratios, none of their values will ever turn out undefined. For example, y/r and x/r are the sine and cosine ratios, respectively, and we know that r always equals one. Thus, these ratios cannot possibly be divided by zero, so they will never be undefined.

However, the cosecant, secant, tangent and cotangent graphs will have asymptotes. This is because their ratios have either x or y as denominators, and there is a possibility of a denominator of zero. Secant and tangent are undefined at pi/2 or 3pi/2 (when x=0), and cotangent and cosecant are undefined at 0 and pi (when y=0).

BQ #2: Unit T Concept Intro

How do the trig graphs relate to the Unit Circle?

1. 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 2pi because, going around the Unit Circle, their patterns take the whole circle to repeat again. For example, sine's pattern is positive positive negative negative. Since there is no clear pattern, we must use the whole Unit Circle to repeat it again and thus, the period is 2pi. The same can be applied to cosine's pattern around the Unit Circle, which is positive negative negative positive. Similarly, it has to be repeated again for a pattern to emerge, so the period for cosine is 2pi (the distance around the Unit Circle). 

Tangent and cotangent, on the other hand, have a clear repeating pattern without having to go all the way around the Unit Circle: positive negative positive negative. We see that "positive, negative" is the repeating pattern and that it repeats itself halfway around the Unit Circle. Then, it starts over again. As a result, the period of tangent and cotangent is only pi, which is half the distance around the Unit Circle.

2. 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 having amplitudes of one relates to the Unit Circle in an important way. First of all, to understand the amplitudes of sine and cosine, we would have to know that the range of the Unit Circle values. They are between 1 and negative 1. And so, with sine having a ratio of y/r--or simply 'y'--the graph's amplitude is limited to one. Thus, the amplitude will remain at one. Cosine, similarly, has a ratio of x/r--or simply 'x', since r is 1--and its graph will have an amplitude of 1.

Reflection #1: Unit Q: Verifying Trig Identities

1. What does it actually mean to verify a trig identity?
To verify a trig identity means to confirm that both sides of the equation are equal to one another, or that the equation is true. We do this by using different identities that we need to prove the equations that we are given.

2. What tips and tricks have you found helpful?
I have found that converting every trig function to sine and cosine helps when verifying or simplifying. Other tricks that I have found helpful is to split fractions that have monomial denominators to simplify even further or to combine fractions with a binomial denominator. It also helps me to look for greatest common factors to factor out of polynomials in the equation and then possibly use Pythagorean identities to simplify.

3. Explain your thought process and steps you take in verifying a trig identity.  Do not use a specific example, but speak in general terms of what you would do no matter what they give you.
My first step is look for any trig functions that can be converted into sine and cosine, as those are the easiest to work with. If the trig functions are squared in the equation, I would also look for the trigonometric pairs that go with each other: sine and cosine, cosecant and cotangent, and tangent and secant. If I have those, then I would use them instead of converting. I might change one of the trig functions to the other using a Pythagorean identity. However, if I do not have any trig functions squared, then converting everything to sine and cosine is the simplest way.

If I have a fraction, I will look for two things. One of them is if I can multiply the numerator and denominator by the denominator's conjugate. This will work only if I have a binomial denominator and if I can't cancel anything out on the top and bottom anymore. After I multiply in the conjugate, I most likely will end up with a Pythagorean identity, so I can substitute in one trig function and split the fraction if I have to--only if I cannot simplify the numerator and denominator of the fraction first.