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.

Both tangent and cotangent's graph direction is due to their asymptotes. Asymptotes lie on the x-axis where x=0. Finding the direction of the graph can be determined by the trig ratio and unit circle.

*color stands for each quadrant in order from 1-4*
(template taken from Unit T Intro Desmo Activity & drawn on by self) 

For tangent, the ratio is y/x. To find the asymptote, x would have to be 0 to be undefined. So when x is 0, we find that the asymptotes lie at (0,1) and (0,-1) which in radians would be pi/2 and 3pi/2. Knowing the asymptotes, we can start on the direction of the graph. Back to the Unit Circle, we know that tangent is only positive in the first and third quadrant and negative in the second and last quadrant. In between the asympotes that we found are the second and third quadrant. Since we know that it's negative in the second, then positive in the third, it goes uphill. The graph will eventually start to repeat itself as the period goes on.

However, for a cotangent graph, it goes down hill. The ratio of cotangent is x/y. Again, asymptotes have a denominator of 0 to make it undefined so we have (1,0) and (-1,0). In radians, it's 0pi and pi. So on our x-axis, these points would be where the asymptote lines are. We know that cotangent is just like tangent in it's positive and negative location. Quandrant I and 3 are positive while quadrant 2 and 4 are negative. We see that the asymptote boundaries are in the first and second quadrant. So it will go from a positive position, down to a negative in the second quadrant. This is how we get our downhill shape.

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.

Sine and cosine are both part of the trig ratio and reciprical identities. All the functions have sin/cos in them so that's how they ultimately relate to each other. When a value in the denominator is equal to zero, we know that is undefined and that is where our asymptote is since asymptotes are undefined. 

Tangent? We know the ratio identitiy is sin/cos. On a Unit Circle, we know that the graph is positive in the first and third quadrant while it is negative in the second and fourth. This leads the the graph to have asymptotes at pi/2 and 3pi/2 since that's where cos=0 and we know anything over 0 is the asymptote. From there, the second quadrant starts with a negative and then in the third quadrant, a positive period. This gives tangent the uphill graph.

Cotangent? As for cotangent, the ratio is opposite of tangent (cos/sin). When sin is 0, we find our asymptotes at 0 and pi on the graph. The period start from being positive in the first quadrant to ending in the negative by the second quadrant on the graph. This gives cotangent its downhill graph. Since both tangent and cotangent only cover up pi units in the x-axis, we see it repeats itself right after an asymptote is drawn.

Secant? Secant is the reciprocal of cosine and has a ratio of 1/cos. So when cos is 0, that's where the asymptote lies. This means it would lie at pi/2 and 3pi/2. In the Unit Circle, we know that secant is only positive in the first and last quadrant. That means in the second and third quadrant, it would be negative. To get from a negative to a positive in an instant, we know that an asymptote will be there when the denominator (cos) is zero. This graph follows the pattern of cosine, however, the differences are the asymptotes.

Cosecant? Cosecant is the reciprical of sine and has a ratio of 1/sin. When sin is equal to zero, we know there will be asymptotes. These would be located at 0 and pi. In the unit circle, we know cosecant is negative in both the first and second quadrant and is positive in both the third and fourth quadrant. This graph follows the pattern as sine, however, the differences are the asymptotes.

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 unlike the other four trig graphs. When we refer back to the Unit Circle and find the ratios, we see that sine has a ratio of y/r and cosine has x/r. We know that an asymptote is undefined so that would only mean the denominator has to be 0. However in this case, our denominator will always be r which will never be 0. That is why sine and cosine do not have asymptotes.

For the other trig functions, tangent, cotangent, cosecant, and secant, these have asymptotes. For tangent and cotangent, we know that it deal with x/y or y/x. It is possible that the denominators of the two can be 0 which will make it undefined and result in an asymptote. As for cosecant (r/y) and secant (r/x), we see that the radius is put as the numerator while x and y are on the denominator. Again, they can value 0 which will make them asymptotes. You can notice that when tangent and cosecant both have y as their numerator they land at (-1, 0) or (1, 0). As for cotangent and secant, they both have x for their denominator and the possible points are at (-1, 0) or (1, 0). This is why the other four trig functions are asymptotes.

BQ#2 – Unit T Concept Intro

How do the trig graphs relate to the unit circle?

Trig graphs relate to the unit circle by the quadants and the positive and negative values. The unit circle is bascially just unraveled onto a line. Since we know that each quadrant as a corresponding value according to it's trig function, the direction of the graph will follow. For example, sine in a Unit Circle is only positive in the first and second quadrant, then in the third and fourth quadrant, it is negative. When it is drawn out on a graph, we know the position of each parts of the line will either be on the positive side or negative. This means that sine will be on the positive the first two parts, then it will be on the negative side. The image below indicates the positive and negative values in each quadrant for each trig function. Following the color coordination and +/- values, the line drawn below shows where each part of the quadrant exists.

(https://docs.google.com/file/d/0B4NSkh2FgPbXQ2p6dmxpdUVLa1k/edit)
*Drawing and labeling done by me*

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

The period of sine and cosine is 2pi because it would take a whole circle to complete the line. We see that both either stay on the positive side or negative side for a good two quadrants. Then, it ends up at where the starting point is. Since the cycle goes through all four quadrants, we know that in a Unit Circle, a 360' would be 2pi. That is why the period for sine and cosine is 2pi. 

The peiod for tangent and cotangent differs. It is just pi. In the image above, it shows how tangent would be graphed on the line. It goes from a positive value, then dips all the way down (representing by an asymptote) towards negative and goes back up the the positive, then back down. From this, we only need to go up to the second quadrant because the cycle repeats itself from there on. In the Unit Circle, we know that the second quadrant goes up to 180', which in radians is just pi. This is why tangent and cotangent's period is pi.

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?

We know that sine has a ratio of y/r and cosine has a ratio of x/r. The "r", or radius, will always equal to 1. Another thing to remember is that sine and cosine can't be greater than 1 or less than -1 or it would be "undefined". So the only way to divide a number from 1 and get either 1 or -1, is 1. This is why sine and cosine have an amplitude of 1.

Unlike sine and cosine, tangent, cotangent, cosecant, and secant don't have an amplitude. These all have ratios of either x/y, y/x, or r's over x and y. There is no limit to them being 1 since the radius is not on the bottom. In tangent, y/x, the value of the varibles can vary and it is not restricted; it doesn't matter whether it equals to  -1/1. That is why these trig functions do not have amplitudes. 

Reflection #1 - Unit Q: Verifying Trig Identities

1. To verify a trig function means to check if the equation is true. To get this, we must get our left side equal to the right side of the equation. Meaning, we already know what the right side is, we just need to get the left side to the same value to verify that the equation is a true statement.
2. The one important tip I found helpful is using your resources. That means all the videos provided, friends around you, and Mrs. Kirch are there to help you in figuring out trig identities. So even if you think you have the answer, your classmates are a big advantage in helping you check if your solution is correct. Another tip that I found helpful is actually memorizing or getting the jist of the types of identities. The ratio, recipricol, and Pythagorean identites all play a significant role and lays a foundation throughout this unit.
3. To verify a trig function, there are many steps that can be taken that can differ from one way to another yet still end up with the same results. The first thing I make sure is that my denominators are the same. If they're not, I have to get them to be the same by multiplying the denominator to the opposing parts or multiplying by the conjugate. Another thing I look at is whether I can convert one thing to a different identity to help me verify/solve the trig identity. You always need to make sure to know which identities can power up or power down or do neither! Itis highly suggested that the trig functions in the identities are similar so that it may be crossed off or simplified easily.