BQ #7: Unit V Concept- Derivatives and The Area Problem

1. Explain in detail where the formula for the difference quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc.) Your post must include text and some sort of media to support your writing.

-The difference quotient is used in higher level math, aka Calculus. In Calculus, the difference quotient is used when we talk about the concept: derivatives. We find the difference quotient by finding f(x+h), the simplifying f(x+h)-f(x) and at the end divide everything by h. When we divide by h we may need to take out a h from the numerator. The difference quotient to us was just this formula in the beginning but now we know that the difference quotient is also known as finding the slope of the tangent line to a graph. So the answer we got from the difference quotient formula is actually the slope of a tangent line aka the derivative. But before we can call it a tangent line we must change it from a secant line to tangent line. The picture below is an example of a secant line because it touches the graph twice.

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We change it from a secant line to a tangent line by plugging in zero whereever there is an h. So all the h's cancel out and makes it a tangent line. Once it is a tangent line we notice in the picture below that a tangent line only tocuhes the graph as we sae above that a secant line touches it twice. Now we have the derivative. 

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BQ #4 Unit T- Concept 3

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

From Mrs. Kirch's SSS packet

tangent: Quadrant 1 is postive, 2 is negative, 3 is postive, and 4 is negative. The ratio for tangent is Tan(x)= y/x. So as you can see when cosine equal zero that means that there is an asymptote. There is an asymptote because when it is zero, it is undefined which means there is an asymptote where ever x equals zero. Tangent has asymptotes at pi/2 and 3pi/2 because that is where x equals zero. So as you can see in the picture above pi/2 is before quadrant 2 which is negative, so the graph will start at the bottom and work its way up because the quadrant before 3pi/2 is postive.

cotangent: The quadrant have the same signs as tangent but the ratio for cotangent is cot(x)=x/y. So as you can see, now it is when sine equals zero where their is an asymptote. There is an asymptote at zero and pi for cotangent because that is where y equals zero. So as you can see in the picture above, the quadrant after 0 is postive and the quadrant before pi is negative. So the graph will start on the top and then go downhill.

So tangent goes uphill and cotangent goes downhill because of the location of their asymptotes.

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.

a. Tangent?

-Sine and cosine relate to cosine because of their signs on the unit cicrle. The ratio for tangent is tan(x)=Sin(x0/cos(x). So since the tangent ratio includes sine and cosine, their signs affect the tangent graph. So if sine is postive and cosine is negative, then the tangent group will be negative and go downhill. If sine and cosine is postive then tangent will be postive and going uphill.

b. Cotangent?

-Cotangent is just the reciporcal of tangent. Cotangent's ratio is cot(x)=cos(x)/sin(x). So cosine and sine's signs depend on what way the graph is going. So if sin is negative and cosine is postive then the graph will go downhill because it will make cotangent negative. We get these signs from the unit circle.  Plus it has diffferent asymptotes than tangent. 

c. Secant? 

-For secant, sine does not affect this graph at all. The only one that affects it is cosine. Cosine affects this graph because the ratio for secant is sec(x)=1/cos(x). So this means that cosine determine where the asymptotes go because cosine can equal 0. If cosine is 0 then it is undefined which means there are asymptotes. So cosine affectets secant because of the asymptotes.

d. Cosecant?

                            

                                     

-For cosecant, cosine does not affect this graph at all. The only one that affects it is sine. Sine affects this graph because the ratio for cosecant is csc(x)=1/sin(x). So this means that sine determines where the asymptotes go because sine can equal 0. If sine is 0 then it is undefined which means there are asymptotes. So sine affects cosecant because of the asymptotes. 

BQ #5: Unit T Concepts 1-3

Mrs. Kirch's SSS packet

5. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use Unit Circle ratios to explain. 

-Sine and Cosine are the only two trig functions that do NOT have asymptotes. The reason behind this is because asymptotes happen when you get undefined. The only way you can get undefined is when you divide by zero. According the the trig ratios, sine is y/r and cosine is x/r. So as you can see, they do not divide by zero because r is equal to one for the Unit Cicrle. Cosecant is r/y and cotangent is x/y so they always have the same asymptotes because sine can equal zero. Secant is r/x and tangent is y/x so they have the same asymptote because they both divide by cosine, and cosine can be zero so that means there is an asymptote present. 

BQ #2- Unit T Concept Intro

From Mrs. Kirch's awesome SSS packet 

1. How do trig graphs relate to the Unit Circle?

-Trig Graphs relate to the Unit Circle because they are basically the same thing. We just unwrap the unit cirlce and make it a line and it turns into a trig graph. It has the same pie values in the same four quadrants. Another reason why they relate is by the signs. The four quadrants stay the same as well. All the signs for each trig functions are the same, so we must remember ALL STUDENTS TAKE CALCULUS. As seen in the picture above you can see how the signs correlate with the unit circle.

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

-The period for sine and cosine is 2pie. The reason behind this is because the pattern for sine is postive postive negative negative. The pattern for cosine is postive negative negative postive. It takes 2pie for this pattern to repeat. So this makes it the period because these graphs go on forever and to repeat the period it takes 2pie. You can see a visual of the sine cosine trig graphs below.

The period for tangent and cotangent is pie. The pattern is postive negative postive negative. As you can see, the pattern repeats half way through the graph/ unit circle so that means that it is only pie and not 2pie because it repeats half way through the revelation. You can see a visial of the tangent trig graph below.

b. Amplitude? How does the fact that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?

-The fact that sine and cosine have amplitudes of one relate to the unit circle big time. On the Unit Circle we remember that sine and cosine could not be smaller than -1 and larger than 1. It relates to this because their amplitudes must be 1 as well. If it is greater than 1 or less than 1 it is considered undefined just like it was on the Unit Circle.

BQ #1: Unit P Concept 2 and 4: Law of Sines SSA and Oblique Triangles

2. Law of Sines- Side Side Angle (SSA) is an ambigious case not like AAS or ASA. When dealing with SSA the three angles are not all known as easily as the others because we only know ONE angle out of the three. It is amigious because it can be three different types: one triangle, two triangles, or no triangle at all.

One Triangle 

As seen in the picture above there is only one possible triangle for this problem. We know there is not a second triangle because angle A and Angle C add up to 325.9 which is way past 180 degrees so therefore there is no second triangle because it is greater than 180.

Two Traingles 

As seen in the picture above this example has two possible triangles. We know there are two triangles because the law of sines means there is one angle in the first quadrant and another angle in the second quadrant. We find the second angle (the prime angle) by subtracting the first angle we got by 180.

No Triangle 

As seen in the picture above this example has no possible triangles. We know there are no possible triangles because once we used the law of sines with SSA, we got sinC: 1.75 and we know from the previous unit that sin can not be greater than 1 so therefore that leads to no solution. Another reason we know a triangle has no solution is when there is more than one obtuse angle.

4. Area Formulas- The area of an oblique triangle is derived from the formula for the area of a trianlge which is A=1/2bh. It area of an oblique triangle is one-half of the product of two sides and the sine of the angle the problem gives you. So basically the three types of equations can be A=1/2bcSinA, A=1/2acSinB, and A=1/2abSinC. It relates to the formula we are familar with by substituting in our h in the normal equation with the a side and sine of an angle given. We just have to make sure when we have our equation, all the letters are different.