Debate Rounds (3)
Anyone got a problem with that?
Similarly the cotangent of x is equal to the cosine of x divided by the sine of x.
AND, EVEN MORE AMAZINGLY, sin (90-x) is equal to cos x!!!!!
Can you tell I'm a student who's actually learning something from his schoolwork?
Feel free to debate that too if you like.
Good luck to my opponent!
Okay. Someone accepted.
Now, most trigonometry problems involve a right triangle like so:
Yes. Now, by definition:
sin(θ) = opp÷hyp
cos(θ) = adj÷hyp
Now, what do we get when we devide sin(θ) by cos(θ), substituting?
We get (opp/hyp) divided by (adj/hyp).
In Algebra, we can multiply (or divide) the numerator and denominator of an expression by the same non-zero quantity without changing the value of the expression.
So, if we divide both the top and the bottom by hyp, then we get opp/adj, which is the definition for tan!!
The cotangent is just the inverse of the tangent, which means it is basically .
So instead of being opp/adj it becomes adj/opp. Which, of course, is what we get when we divide cos(x) (adj/hyp) by sin(x) (opp/hyp), because we can eliminate the hypotenuse just as in the tangent's case.
Did I say anthing else? Oh, yeah. Sin(90-θ) = cos(θ).
This is true because:Sin(90-θ) is actually the sine of the other angle in the triangle. I can prove this as this is a right triangle, which has a 90I0; right angle, and every triangle only has 180I0;. Therefore, the other angle must equal (90-θ).
So, instead of taking the value of sin(θ), we're actually taking the value of sin(k), calling k the other angle. Sin(k) would equal, looking back at our diagram, adj/hyp, as θ's adjacent side is k's opposite side.
And of course adj/hyp constitutes cos(θ).
I don't see how you can counter this, Con, but good luck trying!!
This is where I use the power of...
The most well-known non-euclidian geometry is spherical geometry. It takes place on a sphere.
Here is the triangle which disproves point 1: a triangle who has points on The North Pole, 0"N 0"E, and 0"N 90"E.
This triangle has 3 right angles. From your definition, cos(x) = adj/hyp. Since all three lines are equal (they are all 1/4 the circumference of the Earth), cos(90)=1. However, sin(0) is defined to be 0. If sin(90-x)=cos(x), then 0=1. This is a contradiction, so sin(90-x)=cos(x) is false.
Back to our triangle: from it we can determine sin(90)=1 and cos(90) = 1. But, tan(90) is undefined. Since 1/1 doesn't equal undefined (it equals one), sin(x)/cos(x)=tan(x) is false.
Using the same reasoning as before, you get 1/1 = 0, which means cos(x)/sin(x)=cot(x) is false.
I hope you have a good response to this, Pro.
I was only dealing with plane, Euclidean geometry. I haven't even started the other kind yet.
But Con is quite right. Spherical geometry does work differently.
However, all of my proofs still work, and Con didn't address any of them. Con cannot win just by resorting to equivocation.
Or at least, I'd hope that shouldn't be the case. This is a weird debate.
All you said was, "The tangent of x is equal to the sine of x divided by the cosine of x," and others like that. You didn't address where. I showed that this statement is not true all of the time. This is what is asked of in debates.
In short, I did what is known as winning a debate.
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