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Tutorials / Re: Some Tutorial Links
« on: June 11, 2013, 08:03:34 PM »
Makes life a lot easier! Thank you! I'll be going through these.
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Science / Re: Prove this!
« on: June 09, 2013, 08:43:03 AM »
I got this!
I must admit that I jumped right into the equation and tried to find myself in trigonometric identities, while that was definitely not the way to go.
I did this in a completely rational way, so bear with me. I'm certain there are other ways to solve this, but I felt comfortable following this train of logic... Enjoy pals:
First of all, if you could please direct your attention to the basic, high school Unit Circle:
http://www.regentsprep.org/Regents/math/algtrig/ATT5/600px-Unit_circle_angles_svg.jpg
And then you know that "All Students Take Calculus" way to remember in what specific quadrants are cosine, sine, and tangent negative or positive, because I'm sure your high school teacher kept repeating it the first day of Pre-Calc/Trig class. Here's a reference picture for convenience:
http://www.regentsprep.org/Regents/math/algtrig/ATT3/signchart.gif
Now, the conditions for this proof are that 0<x<pi/2.
This implies that we are trying to prove the statement using only Quadrant I.
I did by proving all Anti-A statements false (while the statement we're proving is A, of course), so I went to the rest of the Quadrants themselves.
In QII, sinxtanx = - n (negative n), because sinx would be positive and tanx would be negative, so +sinx-tanx = -n. However, the right side, which is 2(1-cos), remains positive, because in QII, cosx is negative, so 1 - (-cosx), is equal to something that is greater than 1.
Therefore, -n > +n, is a false statement.
Upon examining the other Quadrants, we get the same statement, -n > +n, which cannot be true.
Therefore, all Anti-A's have been proven false. The only thing that remains is to prove A to be true.
Well, if we are under the condition that 0 < x < pi/2, then we get +n > +n upon the same line of examination (where n just means any random number). So far, so good. Let's go on to the details.
In order that the right side of the statement increases in value, then cosine must be decreasing (because of the 1-cos part of it). As cosine decreasing, it is moving closer to the y-axis, where its value is equal to 0, and where the value of sine is equal to 1. Therefore, the value of sine is also increasing. By extension, the value of tangent (which is just sine/cosine), is also increasing, but by a greater rate than sine, because the cosine value is growing smaller.
Anyway, input some numbers and you get that it never ever goes beyond the value of sinxtanx, so the statement is true and Anti-A's are false. Proven.
TL;DR? Really? I wouldn't have read it either.
I must admit that I jumped right into the equation and tried to find myself in trigonometric identities, while that was definitely not the way to go.
I did this in a completely rational way, so bear with me. I'm certain there are other ways to solve this, but I felt comfortable following this train of logic... Enjoy pals:
First of all, if you could please direct your attention to the basic, high school Unit Circle:
http://www.regentsprep.org/Regents/math/algtrig/ATT5/600px-Unit_circle_angles_svg.jpg
And then you know that "All Students Take Calculus" way to remember in what specific quadrants are cosine, sine, and tangent negative or positive, because I'm sure your high school teacher kept repeating it the first day of Pre-Calc/Trig class. Here's a reference picture for convenience:
http://www.regentsprep.org/Regents/math/algtrig/ATT3/signchart.gif
Now, the conditions for this proof are that 0<x<pi/2.
This implies that we are trying to prove the statement using only Quadrant I.
I did by proving all Anti-A statements false (while the statement we're proving is A, of course), so I went to the rest of the Quadrants themselves.
In QII, sinxtanx = - n (negative n), because sinx would be positive and tanx would be negative, so +sinx-tanx = -n. However, the right side, which is 2(1-cos), remains positive, because in QII, cosx is negative, so 1 - (-cosx), is equal to something that is greater than 1.
Therefore, -n > +n, is a false statement.
Upon examining the other Quadrants, we get the same statement, -n > +n, which cannot be true.
Therefore, all Anti-A's have been proven false. The only thing that remains is to prove A to be true.
Well, if we are under the condition that 0 < x < pi/2, then we get +n > +n upon the same line of examination (where n just means any random number). So far, so good. Let's go on to the details.
In order that the right side of the statement increases in value, then cosine must be decreasing (because of the 1-cos part of it). As cosine decreasing, it is moving closer to the y-axis, where its value is equal to 0, and where the value of sine is equal to 1. Therefore, the value of sine is also increasing. By extension, the value of tangent (which is just sine/cosine), is also increasing, but by a greater rate than sine, because the cosine value is growing smaller.
Anyway, input some numbers and you get that it never ever goes beyond the value of sinxtanx, so the statement is true and Anti-A's are false. Proven.
TL;DR? Really? I wouldn't have read it either.
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General discussion / Re: Those beyond epic videos
« on: June 08, 2013, 05:56:27 PM »
http://www.youtube.com/watch?v=KeJoVeKSsyA "A Fascinatingly Disturbing Thought"
My favorite video. It's more about the thought he discusses itself rather than motivational words, but it all somehow manages to be pretty inspirational at the end.
I loved the video that said we stopped dreaming.
My favorite video. It's more about the thought he discusses itself rather than motivational words, but it all somehow manages to be pretty inspirational at the end.
I loved the video that said we stopped dreaming.
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General discussion / Re: Websites you visit frequently
« on: June 07, 2013, 10:21:10 PM »
This one is the best when I'm being lazy with my free time:
stumbleupon.com
Then there's:
Tumblr
Coursera
Gradelink (My old high school grades. I don't want to visit this website ever again. Ever, ever.)
And maybe some sketchy websites to get my dose of Game of Thrones.
Lastly and least importantly: Facebook, just because I need to pretend I'm a normal person.
EDIT: How could I forget? Collegeconfidential.com
stumbleupon.com
Then there's:
Tumblr
Coursera
Gradelink (My old high school grades. I don't want to visit this website ever again. Ever, ever.)
And maybe some sketchy websites to get my dose of Game of Thrones.
Lastly and least importantly: Facebook, just because I need to pretend I'm a normal person.
EDIT: How could I forget? Collegeconfidential.com
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