Unit R: Student Problem 2

Unit R: Student Problem #3

Unit T Big Question #4

 

 The Fourth Big Question:

Why do sine and cosine don't have asymptotes, but the other four graphs do?

Sine and cosine do not have asymptotes because of their ratios. Cosine is x over r or x over 1. So when a number is input as X, then the value of the ratio will be that number (because the denominator is 1) even if zero was inputted. The same rule applies for sine, which is y/1. So when a value is inputted for Y, the value of the whole thing will be that number.
So what does all this mean?

Sine and Cosine cannot be undefined in these situations. 

Notice all the other trig functions involve cosine or sine being the denominator, and if they end up equaling to zero, then it makes that certain function undefined and creates an asymptote.

Unit T Big Question #3

THE THIRD BIG QUESTION:
Why is a normal tangent graph uphill, but a normal cotangent graph downhill?





The normal tangent graph is uphill for several reasons. Let's start with identities. Tangent equals sine over cosine, and when cosine equals zero, the value becomes undefined. This occurs when the degree value is 90 (pi/2) and 270 (3pi/2). So this means that at these values, there will be asymptotes. Now let's recall ASTC. In the first quadrant, tangent is positive, so the line will begin from the y-axis going upwards (because it's positive) towards the asymptote located at pi/2. The opposite happens in the next quadrant, then the pattern repeats (remember this from previous blog posts). Cotangent is equal to cosine over sine, so let's go with the same logic we used with tangent. If sine were to be 0, then it would become undefined (aka, an asymptote would be present). This occurs at zero degrees, pi, and 2pi.

Unit T Big Question #2

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

Tangent: We graph tangent using cosine to show us where the asymptotes will be located. This is because a tangent line can only be placed when cosine equals 0 (aka, asymptote, because tan=sin/cos)

Cotangent: Graphing cotangent is easy when we have sine because it will show us our asymptotes. For cotangent, sine is the denominator in the ratio so if sine is zero, then there will be an asymptote at that spot for the cotangent graph.

Secant: The secant graph comes from the parent graph of cosine. The cosine graph will show us where the asymptotes of the actual graph will be. Another obvious connection is that secant is the reciprocal of cosine.

Cosecant: Drawing a cosecant graph requires the use of the sine parent graph (the reciprocal). The sine graph will easily show us where the asymptotes of the cosecant graph will be placed, thus showing us how the cosecant graph will look like.

Big Questions Blog Post #1

The BIG Question:

1. How do the trig graphs relate to the Unit Circle?
A Period: Why is the period for sine and cosine 2pi whereas the period for tangent and cotangent is pi?
Periods are cyclical, they are referring to revolutions of the unit circle. A period is only one time around the trig graph (one revolution around the unit circle).




This cycle repeats until the end of time, that's why it is called a cyclical graph. The difference between the periods of sine/cosine and tangent/cotangent is accredited to ASTC. Sine's order is positive positive negative negative. The signs repeat at 2pi. Now if we look at tangent, the ASTC is positive minus positive minus. For this, we only have to go to pi to allow the pattern to repeat itself.


B) Amplitude: 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?

(This is our friend)


Amplitudes are found when sine is pi/2 and 3pi/2, which are the quadrant angles of 90 degrees and 270degrees. Cosine is similar, however this occurs at zero degrees, pi, and 2pi.

The rest of the trigonometric functions (tangent, secant, cosecant, cotangent) are similar. They have asymptotes when the denominator of the ratio is zero (this makes it undefined and it cannot occur).



google images used.

AS #4 Unit S Concept 7, #5

In this video, I give a short walk-through on how to solve a problem from Concept 7. We focus on using half-angle formulas, Pythagorean identities & the Zero Product Property to solve towards the last few steps of the problem. This concept is somewhat of a combination of the last several concepts, so it can be a bit overwhelming, but it is definitely possible.

AS #3 Unit S Concept 3 Number 5

This video is a short lesson on how to do Concept 3 of Unit S, dealing with power-reducing formulas. Our purpose is to reduce the powers of the given problem until they are all to the "first" power. Make sure you deal with the complex fractions correctly and use the correct formulas.

AS # 2 Unit S Concept 4

In these pictures, I solved the same problem twice. However, each time I used different methods: the sum and difference formulas and half-angle formulas. Even though the angles used are not included in the unit circle, this process shows how we can still solve the problem. We know that the answers I ended up with are the same in both instances because when we plug them in, they give the same decimal number.