Goniometrische identiteiten I

A trigonometric identity is an equation involving trigonometric functions that is true for all angles.

One such identity, called the Pythagorean identity, can be derived using a unit circle.

Any point on this circle corresponds to an angle measured from the positive x-axis.

The x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine of the angle.

The x and y coordinates form the perpendicular sides of a right triangle, with the radius as the hypotenuse.

By the Pythagorean Theorem, the square of the y-coordinate plus the square of the x-coordinate equals the square of the radius.

Since the radius is one, this proves that sine squared plus cosine squared equals one.

This is known as the Pythagorean identity and holds for all real angles.

This identity, derived in two dimensions, has applications that extend to three-dimensional systems. In astronomy, celestial coordinates describe the position of stars on the celestial sphere.

The identity ensures that these positions remain consistent on the sphere, allowing astronomers to track stars as they appear to move across the sky.