BQ#4 Unit T Concept 1-3: Graphing all six trig functions

A normal tangent is uphill and a normal cotangent is downhill because of the placement of the asymptotes. For tan it goes uphill because we have asymptotes where cos equals zero(since tan's trig identity is sin/cos and we need to have zero in the denominator to get undefined=asymptotes) and remembering that we need to be in between the asymptotes but not touch them and tan needing to be positive in the first quadrant (because the sin and cos graphs being positive in quadrant one and a positive divided by a positive is positive=tan ) and tan needing to be negative in quadrant two, uphill is the only way you can fit the graph in between the asymptotes. For cot it goes downhill because we have asymptotes when sin equals zero(cot=cos/sin). Cot needs to be positive in the first quadrant(sin and cos are both positive) and negative in quadrant two(sin is postive and cos is negative; positive divided by a negative=negative) and downhill is the only way we can fit the graph in between the asymptotes and it be positive and negative in the second quadrant. To bring it all together the placement of the asymptotes makes a normal tangent uphill and a cotangent downhill.

BQ#3 Unit T Concept 1: Graphing all six trig functions

A.) Once, we draw the sin and cos graphs(sin graph starting and ending at 0 and cos graph starting and ending in its amplitude), we notice that the sin and cos graph are both positive in quadrant one, so the tangent graph is positive because  if tan is sin/cos and sin and cos are positive a positive divided by a positive is positive.In quadrant two, the sin graph is positive and the cos graph is negative, so the tan graph is negative because a positive divided by a negative is negative. In quadrant three, the both the sin and cos graph are negative therefore the tan graph is positive because a negative divided by a negative is a positive. In quadrant four, sin is negative and cos is positive, so the tan graph is going to be negative. Tangent has an asymptote whenever cos equals zero. The asymptotes are determined with sin and cos. Asymptotes happen when we get undefined. We get undefined when we divide by zero or when cos is zero because tan is sin/cos.
B.)We know cot is cos/sin by our trig identities. Looking at quadrant one, we noticed both the sin and cos graphs are positive, so the cot graph will be positive. Cotangent is going to have different locations for asymptotes as opposed to tangent.Cotangent is going to have asymptotes whenever sin equals one. In quadrant two, sin is positive, but cos is negative so the cot graph will be negative. In quadrant three, both sin and cos are negative so cotangent is going to be positive. In quadrant four, cos is positive and sin is negative so cotangent will be negative.The asymptotes are determined with sin and cos.Asymptotes happen when we get undefined. We get undefined when we divide by zero or when sin is zero because cot is cos/sin.
C.)Secant is the reciprocal of cosine. In quadrant one, secant is positive and it goes very high up because if you take the reciprocal of a small fraction you will get a really big number. Sec is going to have asymptotes whenever cos equals zero.In quadrant two, cos is negative so sec is going to be negative because it needs to reflect off from the cos graph. In quadrant three, cos is negative, so sec is negative. In quadrant four, cos is positive so sec is positive.
D.)When sin equals zero there is going to be asymptotes. In quadrant one and two, sin is positive so csc is going to be positive or above the x-axis because it reflects off the sin graph.In quadrant three and four, sin is negative so csc will be negative or below the x-axis. The location of the asymptote shapes our graph. We have asymptotes when we get undefined and  if we look at csc trig identity it is  is 1/sin so sin has to be zero in order for us to get undefined which means asymptotes.

BQ#5 Unit T Concept 1-3: Graphing all six trig functions

To begin with, the Unit Circle ratios for sin is y/r, cos is x/r, csc is r/y( because it is the reciprocal of sine), sec is r/x(because it is the reciprocal of cos), tan is y/x, and cot is x/y (because it is the reciprocal of tan). With that being said, we have amplitudes when we are dividing by zero, which will give us defined because we can't divide by zero. Sin and cos never have asymptotes because if we look at their Unit Circle ratios, r is always in the denominator. As we know when we are dealing with the Unit Circle, r is always 1. Therefore, we never divide by zero, which we do not have amplitudes because it is never undefined because we do not divide by zero.

BQ#2: Unit T Concept Intro: Graphing all six trig functions

Trig graphs relate to the Unit Circle because if we took the Unit Circle and unwrapped it, it will become a straight line and the four quadrants will become hash marks.
A.)So, for sine and cosine the period is 2 pi because sine's pattern is ++-- and cosine's pattern is +--+(based on All Students Take Calculus) and looking at these patterns it will take the whole rotation of the Unit Circle or all of the four quadrants for  the pattern to begin to repeat itself. If we look at the Unit Circle half way at 180 degrees it is pi and at 360 degrees (which is all the way around) it will be 2pi and since it takes sine's and cosine's pattern to begin repeating itself after four quadrants, which is at 360 degrees and 360 degrees is 2pi. One time through their cycle is called a period and therefore sine and cosine have  periods of 2pi. On the other hand, tangent and cotangent have periods of pi(180 degrees) because tan's and cot's pattern is +-+- and as you can see  the pattern is repeated twice. So, after two quadrants(180 degrees) the period starts to repeat itself therefore it is pi, instead of 2pi.
B.)Knowing the fact that sin and cos have to be in between 1 and -1 and that the Unit Circle can extend (0,1 at 90 degrees), (0,-1 at 270 degrees), (1,0 at 0 and 360 degrees), and (-1,0 at 180 degrees) we can state that sine and cosine have amplitudes of one. Also, sin and cos both have r=1 as a denominator.

Reflection#1: Unit Q: Verifying Trig Identities

1.) To verify trig functions it means to prove that the equation on the right is true by showing that the left side equals to the right. In other words, taking an equation and simplifying it until it is exactly the same as the right side. Verifying trig functions can be challenging because there is no set or "correct" method to verify these trig function and all it comes down is to a pick and guess situation and if it takes you to a more difficult path, then you can reverse and try another method.
2.) The number one tip I could give is listen to Mrs. Kirch when she says that you need to study in order to remember all the trig ratios, identities, and Pythagorean identities. The most helpful trick I can give is if you are stuck and you can get things to be converted into sin and cos then you should take that route. Another, tip I can give is to substitute the variables into something that is different from all the sin, pluses, minuses,etc. because a + next to a tan can be confusing.
3.)When verifying trig functions the first thing that I think is can I take out a GCF. If I can then you might want to do that first because perhaps things start cancelling out right there and then. The next step I take is to see if I can substitute an identity. It can be one of the ratio identities, reciprocal identities, or Pythagorean identities. Then, I look if the denominator is a binomial and if it is I can multiply by the conjugate to the numerator and denominator. Another thing you can perform is to combine fractions with binomial denominator or separate fraction, but only when the denominator is a monomial. In addition, you can factor and if all fails you can resort to changing everything to sin and cos.