BQ #3: How do the graphs of sine and cosine relate to eatch of the others?

Sine and Cosine relates to Tangent whenever Cosine/X is equal to zero. It happens to be where the curve on the graph is at. The curve is based on the ASTC of Tangent. Therefore, it will have different curves depending on the ASTC.

Cotangent will have an asymptote whenever sine is equal to zero. When sine is equal to zero, you will have undefined answer, therefore have an asymptote. The graph is uphill/downhill depending on where the asymptote is placed at. In this situtation, asymptote is located at 0, Pi, and 2pi.

Secant is the reciprocal of cosine. Secant will have an asymptote whenever cosine is equal to zero. The asymptote will be at pi/2, 3pi/2. On the first curve, it goes up next to the asymptote because the recicprocal of a fraction will make the number big, therefore will have to increased the curve. The curve will touch the "mountain" based on ASTC however, it will never touch the asymptote.

Cosecant will have an asymptote whenever sine is equal to zero because cosecant is the reciprocal of sine. Sine is zero at 0, pi, 2pi. The curve will depend on the ASTC of Sine. Sine is positive in qudrant 1,2. Because of that, the graph will be above the x asix. In qudrant 3,4 sine is negative. The graph will be below the x axix and will get close to the asymptote.

BQ #5: Why Do Sine and Cosine not have Asymptote, but the other four triq graphs do??

Sine and Cosine not have asymptote because it will never be undefined. The triq function of sine is y/r and cosine x/r. Since r=1, sine and cosine will be never undefined.

 As for the other triq functions, it does have an asymptote because the x, y could equal to zero. Therefore, the triq value of these functions will be undefined. Undefined triq functions will always have an asymptote on the graph.

BQ #2: How do the Triq Graphs Relate to the United Circle?

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

A period is the end and beginning of a cycle. It ends and starts when there is a repeating patterns. In this case. the reason why sine and cosine is 2 pi is because the pattern repeat itself fully around the Unit CIrcle. For tangent/cotangent, the pattern repeats only half way. For example in quadrant 1, tangent is positive, qudrant II negative, qudrant III positive, and quadrant IV negative. Therefore, it repeat itself only half of the graph.

Below are the sign for the triq function. The patterns can be seen by looking at the sign from Qudrant I to Qudrant IV.

The graph below also demonstrates how the triq function of sine and cosine relates to the unit circle. It marks the period of a cycle of a pattern. It would take 2pi for sine and cosine to repeat.

B. How does the fact that sine and cosine have amplitudes of one (and the other triq functions don't have amplitudes) relate to what we know about the Unit Circle?

IT relates to what we  know about the Unit Circle because sine is between 1 and -1 while cosine is between -1 and 1. Therefore, there is no restriction for the other triq function.

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Reflection#1: verifying triq function

24. Verifying is basically like a proof in geometry. You have to prove that the answer is proven true by solving. The differences between verifying identities and proof is that we have to use the identities to prove that something is true. 

25. Some tips and trick I think is help ful include listing all the steps and go through it one by one when you don't know where to start. Also it is important to do it carefully and not make simple mistake such as distributing negative.

26. The first step into verifying a triq function is to look at the problem that they give you. For example,You can tell if you have to foil if the function has a parenthesis. Another would be if you have a fraction. You can find the common denominator and solve. From there you have to use the identity and the basic rule of math to verify your function. The big key here is substitution. You have to use identity to prove/solve the triq function in order to simplify it to 1 function.

SP7: Unit Q Concept 2- Finding Trig Function When Given One Trig Function and Quadrant

This SP7 was made in collaboration with Angela Luong.  Please visit the other awesome posts on their blog by going going here.

I/D #3

Inquiry Activity Summary
1.Today we will be looking at the devriation of the "identity". "Identity" is a proven facts and formula that is always proven true. For example, the pythagorean theoream of a^2+b^2=c^2 is an identity.

To begin with, the pythagorean theorem of a triangle using x, y, and r, is x^2+Y^2=r^2. To make it equal to 1, we would have to divide everything by r^2. Since the ratio for cosine on the  unit circle is x/r and the ratio for sine is y/r, we can say that cos^2theata+sine^2theata=1. Therefore, we can conclude that sin^2x+cos^2x=1 is a pythagorean identity because it is proven to be exactly the same as x/r^2+y/r^2=1. The replacement of x and y with cos and sin proves that it is true.

Another way that is proven true is to plug in the "magic 3" ordered pairs from the unit circle. When we plug in the pairs, we know that it is equal to 1. Therefore, it is an identity.

2. To derive the identity from sin^2+cos^2x=1, first divide everything by cos^2. Sin/tan is equal to tan. Because it is squared with an x, we leave that the same. Cos^2x/cos^2x is equal to 1. 1/cos^2x is equal to sec^2x

To derive it from cosecant and cotangent, you set up the same. Sin2^x+cos^2x=1. To begin with, you divide everything by sin^2x. Sin^2x/sin^2x=1. Cos^2x/sinx^2 is equal to cot^2x. 1/sin^2x is equal to csc^2x.

Inquiry Activity Reflection:

The connections that I see between Units N, O, P, and Q so far are unit circle has relation between these identities. These identities is basically derived from the pythagorean theorem. The pythagorean theorem can also derived from these triq functions.

If I had to describe trigonometry in THREE words, they would be identity, magic ordered pairs, and pythagorean theorem.

WPP #13 & 14: Unit P Concept 6 & 7

This WPP #13-14 was made in collaboration with Angela Luong. Please visit the other awesome posts on their blog by going here

Law of Sines: 

Amber is walking up the street 13 feet towards a store called "Frosty" due North of her car. Her friend sends her a text to get her a cupcake from the store "Cakes-galore!", the shop is located at a bearing of 046*; the "Frosty" store is located at a bearing of 135*. How far is each store from Amber? 

Law of Cosines: 

Amber is walking down to her new job at Clothing Express located 12 feet due North. She hears a cry of "Thief!" and turns her head 32* to see a man carrying a bag of money running 17 feet due East. Amber tries to run after him. How far is the man from Amber?