1-Why do sine and cosine NOT have asymptotes, but the other four trig functions do? Use Unit Cirlce ratios to explain.
The reason why sine and cosine do not have asymptotes is because their trig ratios (sin=y/r and cos=x/r) and always on top of 1 ("r") making them to never be undefined. While the other four trig functions (csc=r/y, sec=r/x, tan=y/x, and cot=x/y) are not over 1 and therefore their denominator can equal 0 which will equal to be undefined and will create the asymptotes.
Thursday, April 17, 2014
BQ #2: Unit T Intro
1-How do the trig graphs relate to the Unit Circle?
A) Period?-Why is the period for sine and cosine 2 pi, whereas the period for tangent and cotangent is pi?
The reason why the period for sine and cosine is 2 pi is because in the unit circle it takes them the whole 360 degrees in order to repeat it's pattern. For example for sine in the first and second quadrants it is positive but in the third and fourth quadrants it is negative so in order to see the pattern repeat one must wait until another 2 pi goes around. While for tangent it only takes pi because in the unit circle it is positive in the first quadrant and negative in the second allowing the pattern to repeat itself again in the next two quadrants. If unclear (for visual) look at the picture below.
B) Amplitude?-How does the fact of that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?
Sine and cosine equaling to one is related to the unit circle because they both can be between 1 and -1 because of their trig functions we used in the Unit Circle before (sin=y/r and cos=x/r). While the other functions don't have amplitudes because of the fact that they don't have restriction. For example, tan=y/x because of this tan can be equal to anything that can be inside and outside the Unit Circle.
A) Period?-Why is the period for sine and cosine 2 pi, whereas the period for tangent and cotangent is pi?
The reason why the period for sine and cosine is 2 pi is because in the unit circle it takes them the whole 360 degrees in order to repeat it's pattern. For example for sine in the first and second quadrants it is positive but in the third and fourth quadrants it is negative so in order to see the pattern repeat one must wait until another 2 pi goes around. While for tangent it only takes pi because in the unit circle it is positive in the first quadrant and negative in the second allowing the pattern to repeat itself again in the next two quadrants. If unclear (for visual) look at the picture below.
Sine and cosine equaling to one is related to the unit circle because they both can be between 1 and -1 because of their trig functions we used in the Unit Circle before (sin=y/r and cos=x/r). While the other functions don't have amplitudes because of the fact that they don't have restriction. For example, tan=y/x because of this tan can be equal to anything that can be inside and outside the Unit Circle.
Thursday, April 3, 2014
Reflection #1: Unit Q: Verifying Trig Identities
1- What does it actually mean to verify a trig identity?
To verify a trig function actually means to solve a very "complicated" equation with trig functions in them. the reason in why i say that they are "complicated" is because in all reality when you know how to do them, the problem will already give you an answer and you just have to see if it is true or you are to just simplify it to the best of your abilities.
2- What tricks and tips have you found helpful?
I found the tip of doing the hardest side first to be very helpful because when I finished the hardest I knew that I was able to solve the rest of the equation fairly quick because I already did the most challenging part of the whole equation. As well was the trick of memorizing the trig identities helped me a lot, yes it was a pain to memorize more equations however in the long run it helped me so much especially during the test because I was able to name identities and was able to cut the work load almost in half. As well as being to split fractions into two monomial denominators because it helped me a lot when I got stuck in solving equations that I thought might have not even had an answer.
3- Explain your thought process and steps you take in verifying a trig identity
Well my first step in doing this is to see whether there is an identity that I can switch for one of the Pythagorean theorems, but if there isn't any then I convert everything to sine and cosine. What I then do is try to cross cancel or I keep looking if there are more Pythagorean theorems. If not and they are in a fraction, then I split them into two monomial denominators or I use their reciprocals to cancel out and find Pythagorean theorems (I only use this when I feel like I am stuck on the problem).
To verify a trig function actually means to solve a very "complicated" equation with trig functions in them. the reason in why i say that they are "complicated" is because in all reality when you know how to do them, the problem will already give you an answer and you just have to see if it is true or you are to just simplify it to the best of your abilities.
2- What tricks and tips have you found helpful?
I found the tip of doing the hardest side first to be very helpful because when I finished the hardest I knew that I was able to solve the rest of the equation fairly quick because I already did the most challenging part of the whole equation. As well was the trick of memorizing the trig identities helped me a lot, yes it was a pain to memorize more equations however in the long run it helped me so much especially during the test because I was able to name identities and was able to cut the work load almost in half. As well as being to split fractions into two monomial denominators because it helped me a lot when I got stuck in solving equations that I thought might have not even had an answer.
3- Explain your thought process and steps you take in verifying a trig identity
Well my first step in doing this is to see whether there is an identity that I can switch for one of the Pythagorean theorems, but if there isn't any then I convert everything to sine and cosine. What I then do is try to cross cancel or I keep looking if there are more Pythagorean theorems. If not and they are in a fraction, then I split them into two monomial denominators or I use their reciprocals to cancel out and find Pythagorean theorems (I only use this when I feel like I am stuck on the problem).
Wednesday, March 26, 2014
SP #7: Unit Q Concept 2- Find all trig functions given one trig function and a quadrant
In this concept we see how we can find all trig functions when given just one trig function and a quadrant. You can find this very easily with SOHCAHTOA or just use this method to check your work. This is the problem of my partner, Genesis R., and mine. Please try to answer it before looking at the answers below:
Tanx= -8/5, Cosx= 5rad89/89
However this isn't the only way to find these answers if you want to learn how to get the same result but with Identities then click here to see how it is done.
Tanx= -8/5, Cosx= 5rad89/89
Wednesday, March 19, 2014
I/D #3: Unit Q- Pythagorean Theorem
Inquiry Activity Summary:
Where does sin^2x+cos^2x=1 come from?
The Pythagorean theorem is derived to become a Pythagorean identity. From the unit circle, the Pythagorean theorem uses "x", "y", and "r" which would become: x^2+y^2=r^2. With this we can perform an operation which allows the equation equal to 1. The operation is to just simply divide r^2 from both sides of the equation which in turn creates (x/r)^2+ (y/r)^2= 1. This then becomes an identity, proven facts and formulas that are always true. From the unit circle we see that cosine equals x/r and sine equals y/r, with these two we can just substitute them into the equation which will then give us the Pythagorean Identity sin^2x+cos^2x=1.
Show and explain how to derive the two remaining Pythagorean Identities from sin^2x+cos^2x=1.
Please look at the pictures below, they already have the descriptions on how I got other two remaining Pythagorean Identities.
Inquiry Activity Reflection:
The connections that I see between Units N, O, P, and Q so far are that they all relate back to the Unit circle. The reason why I say this is because of the fact that when we derive triangles, Heron's Theory, and any of the other equations that we have learned prior to this would reference back to the Unit circle for the legs and hypotenuse which will help us to derive the triangle or theory.
If I had to describe trigonometry in three words, they would be complicated, interesting, and enjoyable. It's complicated because we learn not only the equations and theories given to us but we also learn how it was derived and how it is connected with the real world. It is interesting because prior to learning these mathematical equations and theories you don't really question the math that is going on around you and you especially don't think about how these formulas were derived, so it makes it interesting to see all the math that is around you everyday. And finally it is enjoyable because when you finally understand how to do the problems and understand how you are connected to it, you start to enjoy answering the problem and you begin to see that math isn't as bad as many may think it is.
Where does sin^2x+cos^2x=1 come from?
The Pythagorean theorem is derived to become a Pythagorean identity. From the unit circle, the Pythagorean theorem uses "x", "y", and "r" which would become: x^2+y^2=r^2. With this we can perform an operation which allows the equation equal to 1. The operation is to just simply divide r^2 from both sides of the equation which in turn creates (x/r)^2+ (y/r)^2= 1. This then becomes an identity, proven facts and formulas that are always true. From the unit circle we see that cosine equals x/r and sine equals y/r, with these two we can just substitute them into the equation which will then give us the Pythagorean Identity sin^2x+cos^2x=1.
Show and explain how to derive the two remaining Pythagorean Identities from sin^2x+cos^2x=1.
Please look at the pictures below, they already have the descriptions on how I got other two remaining Pythagorean Identities.
Inquiry Activity Reflection:
The connections that I see between Units N, O, P, and Q so far are that they all relate back to the Unit circle. The reason why I say this is because of the fact that when we derive triangles, Heron's Theory, and any of the other equations that we have learned prior to this would reference back to the Unit circle for the legs and hypotenuse which will help us to derive the triangle or theory.
If I had to describe trigonometry in three words, they would be complicated, interesting, and enjoyable. It's complicated because we learn not only the equations and theories given to us but we also learn how it was derived and how it is connected with the real world. It is interesting because prior to learning these mathematical equations and theories you don't really question the math that is going on around you and you especially don't think about how these formulas were derived, so it makes it interesting to see all the math that is around you everyday. And finally it is enjoyable because when you finally understand how to do the problems and understand how you are connected to it, you start to enjoy answering the problem and you begin to see that math isn't as bad as many may think it is.
Monday, March 17, 2014
WPP 13 & 14: Unit P Concepts 6-7
This WPP 13-14 was made in collaboration with Genesis R.. Please visit the other awesome posts on her blog here.
Hershey has stopped at a stoplight and noticed that his best friend, Marlene, is due west of him at the next stoplight, 30 feet away. Both are going to Hershey's Bakery. Hershey walks N 30* W to get there while Marlene goes N 72* E to get there. What is both of their distances to walk over to Hershey's Bakery?
After having a nice, long conversation with Hershey, they decided to see each other again the next day. They both leave the bakery at the same time. Marlene, in hurry to get to her cousin's Quinceanera, is headed at a bearing of 315* and is traveling 50 MPH. Hershey on the other hand goes home at 30 MPH at a bearing of 078*. How far apart are they after two hours?
Hershey has stopped at a stoplight and noticed that his best friend, Marlene, is due west of him at the next stoplight, 30 feet away. Both are going to Hershey's Bakery. Hershey walks N 30* W to get there while Marlene goes N 72* E to get there. What is both of their distances to walk over to Hershey's Bakery?
Sunday, March 16, 2014
BQ #1- Unit P: Concepts 1&4: Law of Sines AAS or ASA and Area of an Oblique Triangle
i-Law of Sines:
Why do we need it?
The law of sines is of importance because it helps us to find non-right triangles that are commonly seen in the real world. With this we are able to find any unknown angle or side if we have already found or were given two angles and one side as well we could have been given two sides and one angle.
How is it derived from what we already know?
When given an unknown triangle with some given information on the side and two angles, when deriving it we can just form a line straight down from the middle to give us two 90 degree triangles. With this we can use Sine and it will be something as below where we get a ratio. However since we want Sine by itself we multiply through to receive the final answer.
iv-Area Formulas:
How is the "area of an oblique" triangle derived?
When given a triangle with some of the given angles and side we just cut the triangle in half for we can receive two 90 degree triangles. With this we then use SOHCAHTOA to get a ratio, with this we can substitute it in for the height of the area formula we are used to, for we can use to solve the area of the triangle.
How does it relate to the area formula that you are familiar with?
This relates to the area formula (a=1/2bh) that we are familiar with because we solve with this formula however we substitute "h" with 1/2abSinC or any of the others.
References:
http://www.lhs.loganschools.org/~rweeks/trig/law_of_sines.jpg
http://facstaff.gpc.edu/~ahendric/Math1113/sec6_1notes/images/pic010.jpg
http://i1.ytimg.com/vi/Bj7h6OMBvqk/hqdefault.jpg
https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRUfaYjkTy5hlsAuerPsDTfulo_zx2qECBMGIWGwX-EcYy4zuaMlV3LC1XqmMsvMpN1kXNTXUSw9HJScLKdDdZdydgqvLghasCFxtONLYiSbtPDbOBMUe9xWWI1TqvDtxuW0-MR7RxxZY/s400/hi.bmp
Why do we need it?
The law of sines is of importance because it helps us to find non-right triangles that are commonly seen in the real world. With this we are able to find any unknown angle or side if we have already found or were given two angles and one side as well we could have been given two sides and one angle.
How is it derived from what we already know?
 |
| http://i1.ytimg.com/vi/Bj7h6OMBvqk/hqdefault.jpg |
 |
| http://www.lhs.loganschools.org/~rweeks/trig/law_of_sines.jpg |
iv-Area Formulas:
How is the "area of an oblique" triangle derived?
When given a triangle with some of the given angles and side we just cut the triangle in half for we can receive two 90 degree triangles. With this we then use SOHCAHTOA to get a ratio, with this we can substitute it in for the height of the area formula we are used to, for we can use to solve the area of the triangle.
 |
| https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRUfaYjkTy5hlsAuerPsDTfulo_zx2qECBMGIWGwX-EcYy4zuaMlV3LC1XqmMsvMpN1kXNTXUSw9HJScLKdDdZdydgqvLghasCFxtONLYiSbtPDbOBMUe9xWWI1TqvDtxuW0-MR7RxxZY/s400/hi.bmp |
This relates to the area formula (a=1/2bh) that we are familiar with because we solve with this formula however we substitute "h" with 1/2abSinC or any of the others.
References:
http://www.lhs.loganschools.org/~rweeks/trig/law_of_sines.jpg
http://facstaff.gpc.edu/~ahendric/Math1113/sec6_1notes/images/pic010.jpg
http://i1.ytimg.com/vi/Bj7h6OMBvqk/hqdefault.jpg
https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRUfaYjkTy5hlsAuerPsDTfulo_zx2qECBMGIWGwX-EcYy4zuaMlV3LC1XqmMsvMpN1kXNTXUSw9HJScLKdDdZdydgqvLghasCFxtONLYiSbtPDbOBMUe9xWWI1TqvDtxuW0-MR7RxxZY/s400/hi.bmp
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