Pythagorean identity

The Pythagorean trigonometric identity is a trigonometric 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 given by the formula:

(Note that sin²θ means (sin θ)²). This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity.

If the length of the hypotenuse of a right triangle is 1, then the length of either of the legs is the sine of the opposite angle and is also the cosine of the adjacent acute angle. Therefore, this trigonometric identity follows from the Pythagorean theorem.

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 ${\displaystyle x=\cos \theta }$ and ${\displaystyle y=\sin \theta }$ for the unit circle and thus ${\displaystyle x=c\cos \theta }$ and ${\displaystyle y=c\sin \theta }$ for a circle of radius c and reflecting our triangle in the y axis and setting ${\displaystyle a=x}$ and ${\displaystyle b=y}$.

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