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As a consequence of the Pythagorean theorem, the hypotenuse is the longest side of any right triangle; that is, the hypotenuse is longer than either of the triangle's legs. For example, given the length of the legs a = 5 and b = 12, then the sum of the legs squared is (5 × 5) + (12 × 12) = 169, the square 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.
Pythagorean theorem: 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 theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: [6]
A right triangle ABC with its right angle at C, hypotenuse c, and legs a and b,. A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1 ⁄ 4 turn or 90 degrees).
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. 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°).
By the periodicity identities we can say if the formula is true for −π < θ ≤ π then it is true for all real θ. Next we prove the identity in the range π / 2 < θ ≤ π. To do this we let t = θ − π / 2 , t will now be in the range 0 < t ≤ π/2. We can then make use of squared versions of some basic shift identities ...
The formulas show how to transform any right triangle with integer legs into another right triangle with integer legs whose hypotenuse is the square of the first triangle's hypotenuse. A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} .