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

 This has to do with the shift in the asymptotes between tangent and cotangent and the pattern for cotangent and tangent. Cotangent has asymptotes where sine is equal to zero. So its asymptotes are located at 180 degrees(pi) and 360 degrees(2pi). Tangent has asymptotes where cosine is zero. So its asymptotes are located at 90 degrees(pi/2) and 270 degrees(3pi/2). Now we pay close attention to tangent and cotangent's pattern + - + - dealing with ASTC. Once we draw our asymptotes our pattern guides us to whether our graph should be above or below the x axis. If it is positive, it is above the x axis and if it is negative then it is below the x axis. When we draw our graphs, we will see how the placement of the asymptotes affects the shape of the graph, even though they follow the same pattern of + - + -

Why do sine and cosine not have asympotes, but the other four trig graphs do?

Sine and cosine will never have asymptotes because of their trig ratios. The ratio for sin= y/r
and the ratio for cos= x/r. According to the unit circle, r will always equal 1. Asymptotes exist when it is undefined, meaning it has a 0 in the denominator. So this means that sine and cosine will always be over 1, making it impossible for them to have an asymptotes. Tangent,cotangent,cosecant, and secant can have asymptotes due to their trig ratios. Their ratios are as follows:
 tan= y/x
cot= x/y
csc= r/y
sec= r/x
Because they have an x or a y value in their denominator this opens up the possibility to be undefined and therefore have an asymptote.  For example, tangent and secant will have an asymptote wherever the x value equals zero, which is at 90 degrees and 270 degrees. Cotangent and cosecant will have asymptotes where the y value is zero so this is at 0 and 180 degrees.The image below displays the trig functions according to the unit circle.

picture:  http://htmartin.myweb.uga.edu/6190/resources/unitcircletrig.gif

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

Tangent and cotangent have asymptotes where their respective ratios are equal to zero. We know that Tangent=sine/cosine. So when cosine is 0, tangent will have asymptotes. So this means that tangents asymptotes will be at pi/2 (90 degrees) and 3pi/2 (270 degrees). These are the points where cosine, or the x value, is equal to zero. This can also be thought of as cosecant is the reciprocal of sine.  Cosecant is 1/sinx and we know based on the unit circle that sine is 0 at 0 and 180 degrees, so its asymptotes lie there. Whenever we have a zero in the denominator, it is undefined, and undefined means asymptote!  Secant is also the reciprocal of cosine  and its ratio is 1/cosx. Cosine equals 0 at 90 degrees and 270 degrees, so its asymptotes are at pi/2 and and 3pi/2. We know that cotangent=cosine/sine so when sine is 0, cotangent will have asymptotes. So cotangents asymptotes are at 0 and pi(180 degrees), the points where sine is equal to zero. The images below show that cotangents asymptote is in the same place at pi for cosecant.
photos:  http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg94.gif
http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg2.gif

How do trig graphs relate to the unit circle?

Why is the period for sine and cosine 2pi,whereas the period for tangent and cotangent is pi?
 Trigonometric graphs are cynical which means they repeat themselves over and over again. One time through their cycle is called a period. The period for sine and cosine is 2pi while the period for tangent and cotangent is pi.  The pattern for sine is ++-- (meaning it is positive in the first quadrant and second quadrant while being negative in the third and fourth quadrant) This cycle is one period. It takes one full revolution around the unit circle for this cycle to repeat, so one revolution is 360 degrees or 2pi. This is the same for cosine because its pattern is +--+(meaning it is positive in the first quadrant, negative in the second and third quadrant and positive in the fourth quadrant) and it would take one revolution or 2pi for the cycle to repeat. The period for tangent and cotangent is pi. Tangent's cycle is +-+- . This means it is positive in the first quadrant , negative in the second quadrant, positive in the third quadrant, and negative in the fourth quadrant.  Notice that it only takes half the unit circle for its cycle to repeat from positive to negative. Therefore it only takes 180 degrees or pi units to repeat its cycle, making its period pi. The image below gives you a visual of what quadrants the trig functions are positive or negative.

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? 

 The  amplitude is half the distance between the highest and lowest points on the graph.  Sine and cosine have amplitudes of one because based on our knowledge of the unit circle it only goes 1 unit in every direction. So the furthest points it can go to is 1 or -1. So the distance from 1 or -1 to the x axis is 1 unit, making our amplitude 1. Cotangent, tangent, cosecant, and secant do not have amplitudes, however they do have asymptotes. They have asymptotes where sine and cosine equal zero. 

Photo:  http://www.oojih.com/show/trigonometry/generalangle/

Student Video Concept 7 #3

This video explains step by step how to solve an equation with half angle formulas. The problem that we will solve is sin(x/2)+cos(x)-1=0. When solving this problem you need to make sure to check for extraneous solutions. Because we squared both sides, we may have created answers that are not really solutions to the problem. To check for extraneous solutions, plug your answer values back into the original formula, and check if the answer is true.

Student Video Concept 3 #5

This video explains how to use power reducing formulas to reduce our problem so that the problem has the highest power of 1. The problem that we will be reducing is sin^2(x)cos^2(x). Pay close attention when foiling that (2x) is a variable and does not get foiled. Also, pay attention to when we substitute (m) for 2x and know that you will have to plug 2x back in for the value of m. Lastly, you may have to use the power reducing formulas more than once to reduce everything down to the highest power of 1.

Sum and Difference formula in relation to the half angle formula

 The first picture depicts using the half angle formula to find the sine,cosine,and tangent for the angle 105 degrees. The second picture shows using the sum formula to find sine,cosine, and tangent for the same angle. When using the half angle formula, you need to pay attention to what is positive and negative based on the quadrant that your half angle is in. When we solve with the half angle formula and the sum formula we find that our answers are the same for sine, cosine, and tangent. Our answers are written differently when we solve using both formulas, however they are equivalent. We can check by plugging in our answers for sine,cosine,and tangent for both formulas in our calculator , and we find that our answers are the same. The two pictures above depict how to plug our answers into our calculator and shows that they are the same.