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.

Tangent

The graph above shows a picture that has a sine, cosine, and tangent graph all in one. We know that sine is positive in the 1st and 2nd quadrants while cosine is positive in the 1st and 4th quadrants. Apart from this, we also know that the trig ratio for tangent is sine/cosine. So in that case, if both sine and cosine are positive, tangent will be positive. However, if sine is positive while cosine is negative (or vice versa) the tangent graph will be negative which in the end makes the pattern of +-+-. 

Cotangent

The picture above shows a picture that contains sine, cosine, and now cotangent. Cotangent is basically the same thing as tangent in ways that it contains sine and cosine in its trig ratio. However, cosine is now the numerator and sine is the denominator. Again we must follow the ideas of where each trig function is positive or negative depending on the unit circle so we can know what direction to draw it. Apart from this, the asymptotes for a cotangent graph is different than a tangent graph. 

Secant 

The trig function secant , as we know, is the reciprocal of cosine. This makes its trig ratio be 1/cosine. Since we know that it coexists with cosine, we are able to notice the asymptotes so that we can be able to graph the points and make a graph that looks like the blue line above. 

Cosecant 

For a cosecant graph, we know that the reciprocal for this trig function is sine. Since we know that cosecant coexists with sine, we are able to plot the asymptotes (such as secant) and be able to plot the graph which will always look like the black line. 

References 

https://www.desmos.com/calculator/hjts26gwst

BQ #4: Unit T Concepts 1-3

Why is the "normal" tangent graph uphill but a "normal" cotangent graph downhill? Use the unit circle to explain. 

The picture about shows both the tangent and cotangent graphs. The red graph represents a tangent graph while the blue graph shows a cotangent graph. As we can see, both graphs are going in opposite directions. This is because both graphs have different asymptotes which makes the direction of the graphs different. We know that tangent has a trig ratio of sine/cosine. If we make cosine equal 0,  the asymptotes will lie on 90 and 270 degrees. However, using the trig ratio of cotangent, sine of 0 will land on 0 and 180 degrees. Thus, it makes the asymptotes lie on different locations causing a the directions to be opposite.

References:

http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg97.gif

BQ #5 Unit T Concepts 1-3

Why do sine and cosine NOT have asymptotes, but the other trig graphs do? Use unit circle ratios to explain. 

We have learned that trig functions have certain ratios that correspond with them. Sine and cosine do not have asymptotes because because their trig ratios have "r" as their denominator. "R" is always the value of 1 according to the unit circle. This makes it impossible for sine and cosine to have a denominator of 0 which causes an asymptote. Other trig graphs have the possibility of having 0 as the denominator which can create an asymptote. 

BQ #2 Unit T Concept Intro

How do trig functions relate to the Unit Circle?

- Throughout various units, we have witnessed the unit circle have different significances for all units. In this unit we are to notice how the unit circle coexists with trig functions. Trig functions use the unit circle in the way we graph the trig function. We use the unit circle by noticing what quadrants are positive or negative relative to the trig function. For example, sine will be positive in the 1st and 2nd quadrants, so if you unwrap the unit circle,  the graph will be on top of 0 for two quadrants then move t the negative section.

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

-Sine and cosine have a period of 2pi while tangent has a period of pi. This is because sine and cosine require a whole circle rotation to repeat the pattern they have. For example, sine has a pattern of ++--, while cosine has a pattern of +--+. These patterns indicate that it requires a whole unit circle rotation so that it can repeat. Thus making it 2pi. However, tangent has period of pi because it only requires half a circle to repeat it's pattern. It's pattern is +-+-, which indicates that it only takes half a circle to repeat that pattern which makes it have a period of pi.

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 equal to one because they are the only trig functions that have that restriction. The radius of the unit circle is one, so sine and cosine have a denominator of r which is 1. This makes sine and cosine have a radius of one while all the other trig functions do not have a denominator of 1 to depend on.

Reflection #1: Unit Q Concepts 1&5

1. To verify a trig function,  it basically means to make one complex looking trig function into a simpler trig function. When you are "verifying" you will have the final trig function after the equals sign which is your goal to end with after you do different steps.

2. Some tricks or tips that you should use is to basically try to make everything into sine and cosine if possible. If you have everything in sine and cosine, it is an easier passageway to cancel out trig functions, if possible, or create a different trig function using these two trig values. One VERY important tip is to not get frustrated! You will somehow find a way to get through all the steps so be patient!

3. I personally do not have a special process when solving these problems. One step I tend to do is find the best way to start the problem by either multiplying by the conjugate, finding the LCD, splitting the problem, or putting everything into sine and cosine. From there, I search for methods to make trig functions cancel or convert one trig function to another. I continue this process until I finally have my solution.