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Concept
Pythagorean trigonometric identity
Summary
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.
The identity is
As usual, means .
Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless of its actual size: the ratios depend upon the three angles, not the lengths of the sides. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ.
The elementary definitions of the sine and cosine functions in terms of the sides of a right triangle are:
The Pythagorean identity follows by squaring both definitions above, and adding; the left-hand side of the identity then becomes
which by the Pythagorean theorem is equal to 1. This definition is valid for all angles, due to the definition of defining and for the unit circle and thus and for a circle of radius c and reflecting our triangle in the y axis and setting and .
Alternatively, the identities found at Trigonometric symmetry, shifts, and periodicity may be employed. 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 (squaring conveniently removes the minus signs):
All that remains is to prove it for −π < θ < 0; this can be done by squaring the symmetry identities to get
The identities
and
are also called Pythagorean trigonometric identities. If one leg of a right triangle has length 1, then the tangent of the angle adjacent to that leg is the length of the other leg, and the secant of the angle is the length of the hypotenuse.
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