be found along the unit circle as and and By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is Since a rotation of an angle of does not change the position or size of a shape, the points , , , , and are the same for two angles whose difference is an integer multiple of . Thus trigonometric functions are periodic functions with period . That is, the equalities and hold for any angle and any integer . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that is the smallest value for which they are periodic (i.e., is the fundamental period of these functions). However, after a rotation by an angle , the points and already return to their original position, so that the tangent function and the cotangent function have a fundamental period of . That is, the equalities and hold for any angle and any integer . Algebraic values The algebraic expressions for the most important angles are as follows: (zero angle) (right angle) Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. Such simple expressions generally do not exist for other angles which are rational multiples of a right angle. For an angle which, measured in degrees, is a multiple of three, the exact trigonometric values of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by ruler and compass. For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable. For an angle which, expressed in degrees, is a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of th roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic. For an angle which,