Algebra

Algebra:

You can easily remember these values   by writing the numerator as the square root of a continuous non-negative integer with a denominator of 2.

For other angles with a rational multiple of a right angle, this simple expression does not usually exist. For an angle (in degrees) that is a multiple of three, the sine and cosine can be expressed as square roots, see Triangular constants expressed as real roots. These values   of sine and cosine can therefore be constructed from a ruler and a compass.


For integer degrees, sine and cosine can be expressed as square and cube roots of non-real complex numbers. Galois theory allows to prove that if the angle is not a multiple of 3 °, it is unavoidable that the cube root is not real.

For angles expressed in degrees as rational numbers, sine and cosine are algebraic numbers and can be represented by the nth root. This is due to the fact that the Galois group of ring polynomials is cyclic.

For angles measured in degrees that are not rational numbers, the angle or sine and cosine are both priors. This is a corollary to Baker's Theorem, which was proved in 1966.

Simple algebraic values:
The following summarizes the simplest algebraic values  of the trig formulas. The symbol ∞ represents an infinite point on the solid line of the projection extension; it has no sign because when the independent variable approaches the value in the table, when it appears in the table, the corresponding trigonometric function tends to + ∞ on one side, Tends to –∞ on the other side.

ALGEBRA