Pythagoras's theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. That is, for a right triangle with sides of length a, b, and c (where c is the hypotenuse), we have:
c^2 = a^2 + b^2
To prove this using trigonometry, we can use the definitions of sine, cosine, and tangent in a right triangle:
sin(theta) = opposite/hypotenuse
cos(theta) = adjacent/hypotenuse
tan(theta) = opposite/adjacent
First, consider a right triangle ABC with angle A being the right angle, and sides of length a, b, and c as shown in the diagram below.
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A
/|
/ |
c / | b
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/____|
B a
From the definition of sine, we have:
sin(A) = b/c
Rearranging this equation, we get:
b = c*sin(A)
Similarly, from the definition of cosine, we have:
cos(A) = a/c
Rearranging this equation, we get:
a = c*cos(A)
Now, consider the square of the length of the hypotenuse c^2. Using the Pythagorean identity for sine and cosine, we have:
sin^2(A) + cos^2(A) = 1
Multiplying both sides by c^2, we get:
c^2sin^2(A) + c^2cos^2(A) = c^2
Substituting the expressions for a and b that we derived earlier, we get:
(csin(A))^2 + (ccos(A))^2 = c^2
Expanding the squares and simplifying, we get:
c^2*(sin^2(A) + cos^2(A)) = c^2
Using the Pythagorean identity again, we have:
sin^2(A) + cos^2(A) = 1
Substituting this back into the equation above, we get:
c^2 = a^2 + b^2
which is Pythagoras's theorem. Therefore, we have proved Pythagoras's theorem using trigonometry.
Danaus 写了: ↑3月 25, 2023, 12:36 am US teens say they have new proof for 2,000-year-old mathematical theorem
New Orleans students Calcea Johnson and Ne’Kiya Jackson recently presented their findings on the Pythagorean theorem
https://www.theguardian.com/us-news/202 ... etry-prove