Geometric Theorems: Pythagorean & Laws of Sin, Cos, Tangent

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In summary, The Pythagorean theorem is a fundamental mathematical concept that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. The derivation of this theorem has been a topic of much discussion and there are many proofs, one of which involves considering the area of a triangle. The Law of Cosines is a generalization of the Pythagorean theorem, where theta is set to 90 degrees. The original proof of the Pythagorean theorem is believed to have been discovered by Pythagoras himself, although there are many different proofs of this theorem. The proof discussed in this conversation is one that involves using geometry, while other proofs may involve algebra
  • #1
whozum
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Can someone tell me (or help me find) the derivation of the pythagorean theorem, and the laws of sin,cos, and tangent. I know the first is a derivation of the low of cosins, but I'd like to know if there's a writeout as to how he actually came up with those results.
 
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  • #2
There are many proofs to the phytagorean theorem but this is one of my favourite. It was told to me as the applied mathematicians proof of phythagorous. Take a triangle with hypotinuse h, width x, and height y. By a dimension argument the area is [tex]ch^2[/tex] where c is a dimensionless constant. Now draw a line from the right angle that meets the hypotinous at a right angle. Now you have two triangles similar to the original with hypotinuses x and y. So the are of each of these is [tex]cx^2[/tex] and [tex]cy^2[/tex]. And they sum up to the total area so [tex] cx^2 + cy^2 = ch^2 [/tex]

This surely wasn't the original proof but it's very much how an applied mathematician thinks.
 
  • #3
The law of cos is just a generalization of the Pythagorean theorem, let theta = 90. The derivation of the law of cos I don’t remember off hand but it’s just about every trig and pre-cal book.
 
  • #4
Isnt that backwards, jonf?
 
  • #5
whozum said:
Isnt that backwards, jonf?

The Pythagorean theorem long predates the Law of Cosines.
 
  • #6
snoble said:
By a dimension argument the area is [tex]ch^2[/tex] where c is a dimensionless constant.

I'm not following this part, can you elaborate? What 'dimension argument'? Are we drawing any triangle or a right triangle?
 
  • #7
The law of cos is just a generalization of the Pythagorean theorem, let theta = 90

Pythag:

a^2+b^2 = c^2, or is it

a^2 + b^2 -2abcos(t) = c^2?
 
  • #8
whozum said:
I'm not following this part, can you elaborate? What 'dimension argument'? Are we drawing any triangle or a right triangle?


Yeah that is definitely the big jump and this is the sort of stuff some applied mathematicians (especially russian ones) tend to just sweep under the rug. You can do this with any triangle or polygon. Just take any side length and say it is x cm (centimetres). The area will be in cm's squared. So the function between sidelength to area is some multiple of the square of the length since the units have to match and you can't separate the length from the unit. So the constant changes for different triangles, even among right triangles.

Again this is not the sort of proof a Euclidean geometer would come up with. The typical proof is sort of a jig saw puzzle.
 
  • #9
It don't believe one can say HOW Pythagoras himself proved the "Pythagorean theorem" (he certainly did NOT derive it from the cosine law since cosines handn't been invented then!). There are probably more different proofs of the Pythagorean theorem than any other single theorem. Even a president of the United States (James Garfield) developed an original proof- given here: http://jwilson.coe.uga.edu/emt669/Student.Folders/Huberty.Greg/Pythagorean.html
 
  • #10
Thanks ivy that's exactly what I was looking for.
 
  • #11
Here's a diagram for another proof. I'll leave it to you to show the angles work out. Just remember the sum of interior angles is 180degrees.
 

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  • #12
Remeber that back in the days of the Greeks there were no sturctured algerbra like we have now. There proofs were all based of geometry.
 

Related to Geometric Theorems: Pythagorean & Laws of Sin, Cos, Tangent

1. What is the Pythagorean Theorem?

The Pythagorean Theorem is a mathematical theorem that states the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

2. How do you use the Pythagorean Theorem to find the length of a side?

To use the Pythagorean Theorem, you need to know the lengths of two sides of a right triangle. Then, you can square both of those lengths, add them together, and take the square root of the sum to find the length of the hypotenuse. For example, if the two known sides are 3 and 4, the equation would be √(3² + 4²) = √25 = 5. Therefore, the length of the hypotenuse is 5 units.

3. What are the three trigonometric ratios?

The three trigonometric ratios are sine, cosine, and tangent. These ratios are used to find the relationship between the angles and sides of a right triangle. Sine is the ratio of the opposite side over the hypotenuse, cosine is the ratio of the adjacent side over the hypotenuse, and tangent is the ratio of the opposite side over the adjacent side.

4. How do you use the Law of Sines to solve a triangle?

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the opposite angle is the same for all sides and angles. To use this law, you need to know the length of at least two sides and one angle in a triangle. Then, you can use the equation a/sin(A) = b/sin(B) = c/sin(C) to find the missing sides and angles.

5. What is the difference between the Law of Sines and the Law of Cosines?

The Law of Sines and the Law of Cosines are two different trigonometric laws that are used to solve triangles. The main difference is that the Law of Sines is used to solve triangles when you know the length of two sides and one angle, while the Law of Cosines is used when you know the length of all three sides of a triangle. The Law of Sines also involves the sine function, while the Law of Cosines involves the cosine function.

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