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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.
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.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Experts described Jackson and Johnson’s approach as particularly challenging because trigonometry as a field is essentially based on Pythagoras’ theorem; thus using trigonometry to prove the ...
The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
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:
Trigonometry (from Ancient Greek ... The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary ...
— Louisiana students Ne’Kiya Jackson and Calcea Johnson discovered a new way to prove the 2,000-year-old Pythagorean theorem using trigonometry, formerly thought impossible by experts.
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