Exact trigonometric values

In mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 90°. For angles outside of this range, trigonometric values can be found by applying the reflection and shift identities. In the table below, stands for the ratio 1:0. These values can also be considered to be undefined (see division by zero). {| class="wikitable" style="text-align: center;" !Radians!!Degrees!!sin!!cos!!tan!!cot!!sec!!csc |- !!! ||||||||||| |- ! !! ||| || || || || |- ! !! ||| |||||||| |- ! || | || || || || || |- ! !! ||| || || || || |- ! !! ||||||||||| |- ! !! ||| || || || || |- ! !! ||||||||||| |- ! !! ||| || || || || |- ! || | || || || || || |- ! !! ||||||||||| |- ! !! ||| || || || || |- ! !! ||| |||||||| |} Some exact trigonometric values, such as , can be expressed in terms of a combination of arithmetic operations and square roots. Such numbers are called constructible, because one length can be constructed by compass and straightedge from another if and only if the ratio between the two lengths is such a number. However, some trigonometric values, such as , have been proven to not be constructible. The sine and cosine of an angle are constructible if and only if the angle is constructible. If an angle is a rational multiple of pi radians, whether or not it is constructible can be determined as follows. Let the angle be radians, where a and b are relatively prime integers. Then it is constructible if and only if the prime factorization of the denominator, b, consists of any number of Fermat primes, each with an exponent of 1, times any power of two.