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The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae , it is one of the basic relations between the sine and cosine functions.
Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse.
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
The proof is valid for almost all right triangles, except for the case when a=b, i.e. Isosceles Right Triangle. For the case when a ≠ b {\displaystyle a\neq b} , we can always choose to orient the right triangle A B C {\displaystyle \triangle ABC} so that a<b as shown in the figure.
Ne’Kiya Jackson and Calcea Johnson have published a paper on a new way to prove the 2000-year-old Pythagorean theorem. Their work began in a high school math contest.
Two New Orleans high school students have proven the Pythagorean Theorem using trigonometry without relying on circular reasoning. That should be impossible.
When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β. Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas ...
Fig. 6 – A short proof using trigonometry for the case of an acute angle. Using more trigonometry, the law of cosines can be deduced by using the Pythagorean theorem only once. In fact, by using the right triangle on the left hand side of Fig. 6 it can be shown that:
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