Finding the Values of a and b in the Pythagorean Theorem

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Discussion Overview

The discussion revolves around finding the values of the legs a and b in the Pythagorean theorem given the hypotenuse c. Participants explore the conditions under which a and b can be determined, including the necessity of additional information such as angles.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant states that knowing only c is insufficient to determine a and b separately, as there are infinitely many combinations of a and b for a given c depending on the angles of the triangle.
  • Another participant suggests that knowing at least one angle other than the right angle allows for the determination of all three sides using trigonometric functions.
  • It is noted that if the hypotenuse and one leg are known, the other leg can be calculated using the Pythagorean theorem.
  • Several participants emphasize the role of trigonometry in finding the lengths of a and b when the hypotenuse and an angle are known, providing formulas such as a = c sin(angle) and b = c cos(angle).
  • There is a clarification on the notation used for trigonometric functions, confirming the correct representation of the equations.

Areas of Agreement / Disagreement

Participants generally agree that additional information, such as an angle, is necessary to determine the lengths of a and b. However, there is disagreement on the implications of being able to draw a diagram with only the hypotenuse, with some asserting that it leads to multiple possible triangles.

Contextual Notes

Participants discuss the dependency on angles and the conditions under which the Pythagorean theorem applies, highlighting the limitations of the theorem when only the hypotenuse is known.

eNathan
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The pythagorean theorum states that [itex]a^2+b^2=c^2[/itex]. So if you know the value of c^2, or just c, how do you get the vales of a and b, assuming that those arethe legs of a right triangle? And I don't mean the sum of a and b, I mean them seperatly. This has to mathamaticly be posible because you can draw a driagram on paper to do this.

Thanks in advance :-p
 
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You need to know at least one angle other than the right angle, in which case you will know all three angles. Then you can use the definitions of the trigonometric functions to get each side.
 
Yes, you either need to know c and a, or c and b, or c and some angle in addition to the 90 degree angle.

Don't forget, the Pythagorean Theorem only works for right triangles.

For any given value of c there can be infinitely many combinations of values for a and b depending on the angles of the right triangle. But once you chose a value for a or b, then the other is fixed. Keep in mind also that both a and b need to be less than c in all cases.

Alternatively if you know the angles of the right triangle then both a and b are also fixed.
 
"This has to mathamaticly be posible because you can draw a driagram on paper to do this."
No, that's not true.
Given ONLY c, you can draw an infinite number right triangles with that hypotenus- Immagine the hypotenuse pivoting on a point with a "weight" hanging from the other end. As you swing the hypotenuse upward, the horizontal length decreases while the vertical length increases.

Given a hypotenuse length c, a can be any number from 0 to c and then
b= [itex]\sqrt{c^2- a^2}[/itex].
 
Ewo

What I mean is that if you know the hypotenuse, and the angle at which one of the legs IE A and B occur, you can get the value of both of them.

Let me give this example. you have a hypotenuse of 2 meters, and the angle at which B occurs is 40, or something like that, you can draw a diagram on paper to figure out the length of a and b.
 
Last edited:
eNathan said:
What I mean is that if you know the hypotenuse, and the angle at which one of the legs IE A and B occur, you can get the value of both of them.

Let me give this example. you have a hypotenuse of 2 meters, and the angle at which B occurs is 40, or something like that, you can draw a diagram on paper to figure out the length of a and b.

That's what trigonometry is for.

b = c cos40
a = c sin40
 
Gokul43201 said:
That's what trigonometry is for.

b = c cos40
a = c sin40

Does that mean
b = c*cos(40)
a = c*sin(40)

:confused:
 
Yes, that's what he wrote, that's what he meant
 

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