 The primary solution angles on the unit circle are at multiples of 30 and 45 degrees.

Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification.

All values of sine, cosine, and tangent of angles with 3° increments are derivable using identities: Half-angle, Double-angle, Addition/subtraction and values for 0°, 30°, 36°, and 45°. Note that 1° = radians.

This article is incomplete in at least two senses. First, it is always possible to apply a half-angle formula and find an exact expression for the cosine of 1/2 the smallest angle on the list. Second, this article exploits only the first two of five known Fermat primes: 3 and 5; and the trigonometric functions of other angles, such as (= 40°), and (as well as the other constructible polygons,  , or are soluble by radicals. In practice, all values of sine, cosine, and tangent not found in this article are approximated using the techniques described at Generating trigonometric tables.

The values of sine, cosine, and tangent of angles with 1° increments can also be derived using the triple-angle identity. However, they can only be expressed with intermediate (and irreductible) complex numbers in the expression (see the formulae to compute the roots of a cubic equation), or by transforming them using hyperbolic real functions).

## Table of constants

Values outside [0°, 45°] angle range are trivially extracted from circle axis reflection symmetry from these values. (See Trigonometric identity)

### 0°: fundamental    ### 3°: 60-sided polygon    ### 6°: 30-sided polygon    ### 9°: 20-sided polygon    ### 12°: 15-sided polygon    ### 15°: dodecagon    ### 18°: decagon    ### 21°: sum 9° + 12°    ### 22.5°: octagon    ### 24°: sum 12° + 12°    ### 27°: sum 12° + 15°    ### 30°: hexagon    ### 33°: sum 15° + 18°    ### 36°: pentagon    ### 39°: sum 18° + 21°    ### 42°: sum 21° + 21°    ### 45°: square    ### 60°: triangle    