WPP #10: Unit L Concept 9-14

                                     

SV #4: Unit I Concept 2: Solving Logarithmic Equations

**to watch my video click here

For this concept when you're finding the h and the k, the h will be coming from the number that is inside the parenthesis in the equation and it will be opposite sign of what is inside. The k will be coming from the last number in the equation. The range in this case will always be from negative infinity to positive infinity. The domain will be from whatever you h is to infinity. For the x-intercepts you have to put y=0, for the y-intercept you have to put x=0.

SV #3: Unit H Concept 7: Finding Logs with Given Approximations

**to watch my video click Here

For this concept it is important that you realize that since you are putting the simplified numbers in a fraction you need to put the subtraction sign I between separating the tops from the bottom. Also try to remember that if you have log(base4)4 the your answer will be 1 because 4 to the power of 1 is 1 . Also if it is log(base4)1 the that will equal 0 because anything to the power of 0 is 1.. If you use one of those two don't forget to add it to the end of your answer.

BQ #4: Unit T Concept 3

Why is a “normal” tangent graph uphill, but a “normal” Cotangent graph downhill? Use unit circle ratios to explain.

In tangent's ratio in the unit circle it's y/x. When you're trying to find the asymptotes for it you have to look where x would be undefined which means that the x has to equal 0. You would have to find the ordered pairs in which 0 is in the x spot, that would be in the 90 & 270 degrees. When you're going to put the asymptotes it would be on pi/2 & 3pi/2. Therefore the asymptotes would be at the 2 & 3 "quadrant", that will make the graph start from the bottom and make its way to the top without touching the asymptotes, being an uphill graph.
                                 
For cotangent the signs on the graph would be the same but the only thing that will make the difference would be the ratio, you now have the ratio of x/y. That means that you have to find the degrees in he unit circle where y will be 0. That could be found in 180 & 360 degrees. The asymptotes will shift from what they were in tangent. The asymptotes would be from the 3rd and 4th quadrants which will start of with the positive and then go down to negative. The graph will now be a downhill graph.

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.

Tangent

                                

When you look at the unit circle you realize that tangent in positive in 1st and 3rd quadrant and negative in the 2nd and 4th quadrants, so when you unfold the unit circle in the graph you divide it by quadrants. To mark the quadrants you label them with the radians of the quadrants so it's pi/2, pi, 3pi/2, and 2pi, so since tangent is positive in the 1st & 3rd quadrant they would have asymptotes at pi/2 and 3pi/2.. In the graph the second "quadrant" is negative while the third quadrant is positive, the graph has to stay in between the two asymptotes so the tangent graph will start in the negative and go up to the positive without touching the asymptotes. 

Cotangent

                                 

For cotangent everything is mostly the same but the ration is the inverse of tangent,which makes a huge difference in the graph. The ratio is now x/y which we now have to find where x=0, x is 0 at pi and 2 pi. The asymptotes for cotangent will now lie at pi and two pie which means that they just shifted, when they shifted the signs now start from positive to negative which will make the graph downhill.

Secant 

Secant, just like cosine, is positive in the 1st and 4th quadrants and negative in the 2nd and 3rd quadrants. The asymptotes for this graph will lie on pi/2 and 3pi/2. Therefore that means the in quadrant 1&4 the parabolas will be above the positive cosine but with the inverse direction. Now in quadrants 2&3 will be below the positive cosine and will also have the inverse direction. 

Cosecant 

For cosecant, just like sine, it will be positive in the 1st and 2nd quadrant and negative in the 3rd and 4th quadrant. The asymptotes for cosecant would be on pi and 2pi since you have to find where y=0. That means that in the 1&2 quadrant it will have a parabola at the peak of the amplitudes with an inverse direction. Then in the 3&4 quadrant it will be negative  and heading at an inverse direction of  the amplitude. It would be very similar to the sine graph. 

BQ #5- Unit T: Concepts 1-3

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

When you look back at the unit circle for SINE, COSINE, and TANGENT we had ratios to which we derived the SOH-CAH-TOA where we could see the where the ratios came from. When you see sine and cosines ratio they are both over "r". For sine it's ratio is y/r and then for cosine it is x/r. In the unit circle, it's whole radius is equal to one, therefore that means that sine and cosine will not have asymptote because hey are just over 1 unlike all the other functions. Cosecant, secant, tangent, and cotangent are all over different numbers which makes them have an asymptote, the asymptote being the number the equals either the x or the y. In cosecant and secant their ratio has the r on the top of the fraction leaving the bottom with other variables. 

WPP #9- Unit L Concepts 4-8

SP #6- Unit K Concept 10: Writing a repeated decimal as a rational number using geometric series

For this concept you have to pay most attention to where the numbers come from and why he have the decimal where they do. Like the .18 comes from 4.181818 and it is put over 100 because it's in the hundredths place. The 100 goes down to 99 because you had to subtract 1 from it like its given to us in the equation.

SP #3- Unit I Concept 1: Graphing Exponential Functions

When graphing exponential functions you have to make sure that when you are trying to find the x intercept and you have to use the natural log you have to remember that you can't take the natural log of a negative number or fraction. To find the x intercept you have to make y=0 and to find the y intercept you have to make x=0. The domain of the graphs will always be from negative infinity to positive infinity.

BQ#2- Unit T Concept Intro

How do the trig graphs relate to the unit circle?

The trig graphs are just the unit circle unraveled. If you were to take the circle and stretch it into a line then you get your four quadrants but just in a straight line. In the unit circle each quadrant angle could be turned into radians, the first quadrant would be pi/2, the second quadrant would be pi, the third quadrant would be 3pi/2, and the fourth quadrant would be 2pi. For sine the pattern would go POSITIVE for the first two quadrants and NEGATIVE for the last two quadrants. In cosine the pattern is POSITIVE, NEGATIVE, NEGATIVE, POSITIVE. Finally for tangent it will start of with POSITIVE,NEGATIVE, POSITIVE, and NEGATIVE. as you see the "All Students Take Calculus" also applies to the graphs. 

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 2pi because unlike tangent and cotangent the pattern for sine and cosine does not repeat until the second time around the "unit circle". The patterns aren't repeated which means that for a full period it has to reach 2pi. For tangent and cotangent it's period is only pi because it's pattern is repeated (positive,negative,positive,negative) which means that there is no need to reach 2pi because it would be exactly the same. 

Amplitude?- how does the fact that sine and cosine have amplitudes of one (and ogre trig functions don't have amplitudes) relates to what we know about the unit circle?

Sine and cosine both have the ratio of having "r" as the denominator. Cosine's ratio being x/r and sine being y/r, in the unit circle "r" always equals one since it is the radius of the unit circle.  With that you realize that the boundaries for sine and cosine is between 1 and -1. Now for tangent it's different because the ratio for tangent is y/x and the x can be multiple values not only 1 and -1 which means it has no restrictions. 

Reflection #1: Unit Q: Verifying Trig Functions

1. What does it mean to actually verify trig functions?

In order to verify a trig function you have to make sure that the final product is the same on both sides. They start of by giving you a complex problem on one side of the equal sign and on the other side is the simplified version of it. You have to continue by trying to find identities or ways to simplify the equation on the left to equal the one on the right. If you want to use an identity you have to make sure that it is actually and identity. At the end both side need to equal the same and if they don't then you will have to try to solve for it again.

2. What tips and tricks have you found helpful?

For concept 5 I found out that the tricks is that you first have to see if you can find the least common denominator or try to separate the equations if there is one identity at the bottom and if all odds fail then you will have to turn whatever you have into sine and cosine. There are times in which you can't tell but you are able to cancel things out or you are able to substitute it for an identity. You can try to simplify. When you have to do the conjugate you have to make sure that you distribute and factor it out correctly.

3. Explain your thought process and steps you take in verifying a trig function?

For me verifying trig functions is extremely hard for me but I just try to find any identities but then if I can't then I have to see if I can either separate the functions or try to use the conjugate. Once I find anything then I try to use my identities and canceling things out or trying to simplify it.

I/D #1:Unit Q-Pythagorian Identities

Inquiry Activity Summary

1. Where does where sin2x+cos2x=1 come from to begin with?

• The equation sin2x+cos2x=1 derives from the Pythagorean Theorem. As you can see if you were to replace both sine and cosine with the variables with variables you can get the regular a^2+b^2=c^2 which is the equation that we usually use to find a side of a right side triangle. Or then we can also replace it with x^2+y^2=r^2 which is the equation that we have been using to solve some triangles in unit P, but for this unit we want the "r^2" to be equal to 1. To be able to do that you would have to divide the equation by "r" which will give you ratios from the unit circle that you can be able to change to cosine, sine, or tangent depending on the ratio. So you will get (x/r)^2+(y/r)^2=1, the (x/r) is the ratio for sine and (y/r) would be the ratio for cosine. The new equation that you would get would be sin2x+cos2x=1. 
• 30 degrees= (radical 3 over 2, 1/2)
• 45 degrees= (radical 2 over 2, radical 2 over 2)
• 60 degrees= (1/2, radical 3 over 2)

2. Show and explain how to derive the two remaining Pythagorean Identities from sin2x+cos2x=1.

 

Inquiry Activity Reflection

1.The connections that I see between Units N, O, P, and Q so far are that all of the units so far is at all of the sections begin to piece back to the unit circle from it being the angles or the ratios or the plotting points. Then there's the use of equations except for the fact that we just begin to change variables around to fit he concept. 

2. If I had to describe trigonometry in THREE words, they would be PRACTICE MAKES PERFECT.

WPP #13&14: unit P Concept 6 & 7- Applications with Law of Sine and Cosine

       Jaime's and Sol's unfortunate trip to Juana's Bakery

This WPP #13 & 14 was made in collaboration with Sol S. Please visit other rad posts on his blog by clicking Here!!!!!

LAW OF SINE'S

Here is our story:
Sol and Jamie are eating at Juana's Bakery at two separate locations 9 miles apart. Both have eaten tainted bread and need to be rushed to the hospital. Sol is rushed North 30 degrees east from where he is and Jamie is rushed north 42 degrees west from where she is. Find the missing distances they were both rushed. 

 

After working out through the problem of sines we found out that Sol was 7.03 miles away and Jaime was 8.2 miles away from the hospital. Therefore Jaime was probably dying from Juana's Bakery's food poisoning. 

LAW OF COSINE'S

Both Jaime and Sol leave the hospital with a horrible stomach ache/problem at 5pm. Sol is headed to his house at a bearing of 088 degrees and traveling at 14.5 miles per hour. Jaime is traveling at 9 miles per hour at a bearing of 43 degrees. How far apart are they when they have to take their medicine at 9pm?

Once you complete the law of cosines you figure out that Jaime and Sol were 41.3 miles apart when the time to take there medicine came!!

WPP #12 Unit O Concept 10: Solving Anlges of Elevation and Depression

Create your own Playlist on LessonPaths!

ID#2 Unit O Concept 7-8: How can we derive the patterns for our special right triangles?

Inquiry Activity Summary

       In the activity that we were given in class we were supposed to derive a 45-45-90 out of a square making sure that both of the triangles that we make produce a 45-45-90 triangle.  All of the sides of the square were equaling to 1 and we had to find the value of the sided and the hypotenuse, making sure that it will still be equal to 1. Then to form the 30-60-90 we were given an equilateral triangle in which all the sides were also equaling 1. For the 30-60-90 we also had to find the hypotenuse and the two sides all equaling to 1 at the end.

45-45-90

       For the 45-45-90 we are given a square in which obviously all sides are equal but they are equal to 1 and we have to make a 45-45-90 triangle out of it. I found the 45-45-90 triangle by dividing the square diagonally to split the 90 degrees into 45 degrees leaving the triangle with only one 90 and two 45 degree angles. From there we have to find out how to make all the sides still equal to 1 even when it has been divided. what is missing this time is the hypotenuse which means that you have to do the Pythagorean to find the hypotenuse and since each side is still 1 all you have to do is square 1 which is still one and add them which will equal to 2 and from there you have to square root it and that will be rad. 2. the "n' was being used to be able to make the triangles in different sizes using "n" as the ratio in which they can be interchangeable.

30-60-90

      For the 30-60-90 we were given an equilateral triangle in which had to set up the triangle to be able to make two 30-60-90 triangles to have all sides still equal 1. I divided the triangle straight down the middle to form the two triangles, if you divide it down the middle you get the 30 degrees at the top since you split 60 in half, the 60 at the side, and the 90 down the middle since you divided 180 by two. Once you have divided it by the middle you now have the sides equal 1 and the bottom both equal 1/2. To be able to get the variables to equal to "n", "2n", and "n rad. 3" you have to do the Pythagorean to find you b since you already have your hypotenuse and one side you plug in what you know. once you have completed the Pythagorean you will know the sides but since it is in fractions you can make it into whole numbers by multiplying my two to get rid of the 2 at the bottom. Using "n" allows for there to be a ratio in the equation therefore being able to enlarge or to shrink the triangle.

Inquiry Activity Reflection

Something i never noticed before about special right triangles is that they derive from a square or an equilateral triangle and that how the variable derive from.

Being able to derive these patterns myself aids in my learning because now i further understand where the equations come from, they make sense as into form where the "n" comes from and why it has the radicals or other numbers in front of them.

RWA#1 Unit M Concept 4-6 Graphing and Identifying Conic Sections

                                     Lets Get to Know Ellipses

1. Mathematically Definition: The set of all points such that the sum of the distance from the two points known as the foci is a constant.(Mrs. Kirch)

2.Describing the Conic Section

Algebraically:

                    

             Algebraically the equation of an ellipse will determine the fact whether it will be skinny or if it will be fat. In the equation it is easy to identify if its skinny or if it will be fat, if its skinny the number under the "x" would be greater and if it is fat then the number under the "y" would be greater. Also the "h" and "k" identify the center of the graph but once you have identified the center in the equation then you just have to switch the signs of the numbers in the "h" and "k" spots. You then also have to remember that "x" always goes with 'h" and "y" always goes with "k". then also remember that the ellipse is the one that has the plus sign inbetween the equations.

Graphically:

   

                 

         To begin to form the graph for your equation you first have to make sure that the equation is in standard form and if it inst then you have to use completing the square. Once you have completed the square and your equation is now in standard form you then continue by identifying the center (don't forget to switch the signs) and plot the point once you have identified it. You then can continue by finding the major and minor axis, the major axis would be the one that has the bigger degree on the bottom and the minor will have the smaller degree. You can continue by finding "a" and "b" those can be the square roots of the numbers on the bottom of the equation. From that you do a^2-b^2=c^2 and that will give you the c. The 2 vertices can be found depending on your major axis that number can be substituted by that number and the other number will be the difference of the center. The 2 co-vertices will have the minor number and then the difference of the numbers. For more information click here

3.Real World Application

         Ellipses are extremely common in our real world because they can be found in the earths orbit around the sun, its an ellipse because it is almost a perfect circle but not exactly which therefore makes it an ellipse. The earths eccentricity is a 0.0167 which would make it a  skinny ellipse compared to that of the orbit of Pluto which would turn out  to be 0.2488. The focus of the Earths orbit would be the sun. Since it is not the center of the orbit of the sun therefore that means that the earth can either move further away or closer throughout the orbit. For more information click here.

4. Work Cited

1. https://www.youtube.com/watch?v=5nxT6LQhXLM
2. http://www.physics.unlv.edu/~jeffery/astro/ellipse/ellipse_001.png
3. http://mathworld.wolfram.com/Ellipse.html
4. http://www.windows2universe.org/physical_science/physics/mechanics/orbit/ellipse.html
5. http://www.agt.net/public/garnold/ellipse2.gif