Engineering Thermodynamics by J. Pohl © www.thermomentor.com

The Pythagorean Theorem is about the sides and hypotenuse of right triangles. A large area equals the sum of its parts. This idea results in the theorem. A step further - definition of the ratio of sides - yields its sine of angle and cosine of angle form.

Theorem of Pythagoras image The underlying principle of this famous theorem is that an area equals the sum of its parts. In this case the areas are squares and triangles. The angle, θ and the sine and cosine functions are defined.

Prove both versions of the Pythagorean Theorem.

A2 + B2 = C2      and         sin2θ + cos2θ = 1

♦  By the sketch, we see the area of the circumscribed square is:

(A + B)2

And by a previous proof, this equals:

A2 + 2AB + B2.

The area of the large square equals the area of the smaller square: C2, plus four triangular areas: 4 (AB/2).

A2 + 2AB + B2 = C2 + 4 (AB/2)

Thus we obtain the PYTHAGOREAN THEOREM:

A2 + B2 = C2.

To proceed a step further, divide the theorem by C2, to obtain:

(A/C)2 + (B/C)2 = 1

But the ratio (A/C) is named the "sine" of the angle opposite side A. And the ratio (B/C) is the defined to be the cosine of the angle adjacent to side B and we obtain:

sin2θ + cos2θ = 1