Big Question Blog Post # 3

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

The way that the tangent or cotangent graph faces has to do with the asymptotes and with the quadrants where tan/cot is positive or negative in. A normal tangent graph has asymptotes where it is undefined or where x is equal to zero, this will be at 90 degrees and at 270 degrees. After we figure out where our asymptotes are, we need to look at the positive and negative signs and we can draw the curves from there. The pattern for tangent is + - + - so first we draw the graph going upward, then downward, then upward,and  downward again.

A normal cotangent graph goes downhill because we know that cot= x/y. There will be asymptotes where y equals zero and that will be at zero degrees and at 180 degrees.  Next, we need to look at the pattern again, first it is positive, then negative. When we draw the positive side, we need to draw it above the x-axis and when we draw the negative side, we need to draw it below the x-axis.We need to start drawing our graph at the nearest asymptote so that is can go downhill That is how we know that cotangent goes downhill.

Big Question Blog Post # 4

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

Sine and cosine will never have asymptotes. Sine does not have asymptotes because sine is equal to y/r. We know that "r" can never equal zero and in order for us to have asymptotes we need to have zero in the denominator because asymptotes only have when something is undefined. Cosine is equal to y/r and since we know that "r" can't be equal to zero, it can't have any asymptotes.

Tangent and Cotangent have asymptotes because tangent equals  y/x and cotangent equals x/y. Tangent and cotangent can have asymptotes because based on the unit circle, there are some angles there where x is zero or y is zero. Remember, asymptotes are formed when we have an undefined value or any number divided by zero. We can have asymptotes in tangent and cotangent because the quadrant angles have zeros in their ordered pairs and those zeroes give us asymptotes. 

Big Question Blog Post #2

How do the graphs of sine and cosine relate to the others? Emphasize asymptotes in your response.
a. Tangent: The sine and cosine graphs relate to the tangent graphs because we need to go through the same process to graph these trig functions. The sine and cosine graphs relate to the tangent graphs because tangent equal sine/cosine, so if we were to graph sine and cosine on the same graph, we would know where to graph tangent. We also know that since tangent equals sine/cosine we will have an asymptote where cosine is equal to zero.

b. Cotangent: Sine and cosine graphs relate to tangent graphs because we know that cotangent equals cosine/sine. When we graph cotangent, we know that we will have asymptotes where sine is equal to zero. The only reason that we have asymptotes is because we have a value divided by zero so it makes it undefined and we get an asymptote.

c. Secant: The sine and cosine graphs relate to secant because secant is the reciprocal of cosine. When we graph secant, we sketch the cosine graph first. The reason that we sketch the cosine graph first id so that we can now where we need to draw the asymptotes and the little parabolas that go on the graphs. Secant is related to sine and cosine because secant is the reciprocal of cosine.

d. Cosecant: Cosecant is related to sine and cosine because cosecant is the reciprocal of sine. When we graph cosecant, we start with the same process for graphing sine. We need to lightly sketch the sine graph because it will be our guide to draw our cosecant graph. Then we draw the shifts and draw the parabolas where those shifts are located. Cosecant is related to the seine and cosine graphs because cosecant is the reciprocal of sine.

Big Question Blog Post #1

How do the trig graphs relate to the Unit Circle?

The trig graphs relate to the unit circle because when we graph a trig function, we are basically taking the unit circle and stretching it into a line. We also need to look at ASTC to know how the trig graphs relate to the unit circle. Sine's pattern is positive, positive, negative, negative. That means that in the first two quadrants sine is positive and in the next two quadrants, sine is negative. When we graph sine, the first two sections are above the x-axis because they are positive. The next two sections of the sine graph are below the x-axis because sine is negative in quadrants three and four. In the picture below, we see how the sine graph relates  to the unit circle because we know that the sine of 90 is one and that is shown on the graph. The cosine and tangent graphs also relate to the unit circle based on the pattern they have. For example, cosine's pattern is positive, negative, negative, positive. Tangent's pattern is positive, negative, positive, negative. 

a. Period: The period for sine and cosine is 2 pi and the period of tangent and cotangent is pi. The period for sine is 2 pi because the pattern for sine and cosine takes two periods to to repeat. The pattern for sine is + + - - and the period for cosine is + - - +. The period for sine and cosine because their respective patterns take two periods to repeat. The period for tangent and cotangent is only one period long because the pattern only takes one period to repeat itself. The pattern for tangent is + - + -.

b. Amplitude: Amplitude is the distance between the lowest point and the highest point on the graph. We know that sine and cosine have amplitudes because of the values on the unit circle. Tangent and cotangent don't have amplitudes because those graphs go uphill and downhill. Since these graphs go uphill and downhill, it is not necessary to have an amplitude.

Assessment # 4

1. What is this problem about?

This problem is from Unit S Concept 7. Concept 7 is about solving equations with half-angle formulas. This video will go over number 3 from the SSS.

2. What does the viewer need to pay attention to in order to do the problem correctly?

The first thing that we need to do is replace the half angle of sine with its half angle formula. There is a radical in the formula, so later in the problem we square both sides to get rid of the radical. When we square both sides we need to remember to check for extraneous solutions. The most important thing about this problem is to check for extraneous solutions because some of the answers we get may be wrong.

Assessment #3

1. What is this problem about?
This is a problem from Unit S Concept 3 Number 1 from the SSS. Concept 3 was about using the power reducing formulas. For this concept, the viewer has to keep in mind that we want the highest power to be one.

2. What does the viewer have to pay attention to in order to do the problem correctly?
The viewer needs to remember that the highest power has to be one, so they need to reduce everything else that is bigger than one. The viewer also needs to follow each step because these problems can get tricky and the process is kind of long.

Assessment #2

I chose to work with the angle 105.

In Unit R, we solved it with the sum and difference formulas

In Unit S, we learned how  to solve it with half angles

In both pictures, the same angles was used in the problems. In the first picture  we solved by using the sum and difference formulas. This means that we used two angles that added up to 105 degrees, in this case the angles were 60 and 45. then we used the ordered pairs from the unit circle and plugged them into the formulas to solve. In Unit S, we learned about half-angles. Basically, we can solve for an angle that is not on the unit circle by using an angle that is on the unit circle and diving it by two. Then we use the ordered pair from the unit circle and plug it into the formulas to solve. We know that these answers are the same because we used the same angle in both of the problems. In one of the pictures, we added two angles to make 105 and in the second picture  we divided an angle by two to get 105. Since we used values from the unit circle, the answers were the same because the ordered pairs on the unit circle are similar to each other. To prove that the answers in both of the pictures are the same, we can plug them into our calculator and compare the decimal answers.