exact trigonometry tables

0.1 Basic angles

Since the trigonometric ratios for most angles cannot be calculated exactly in closed algebraic form, a few well-known angles that can be calculated often comprise the bulk of textbook exercises involving trigonometryMathworldPlanetmath.

The basic angles are given in Table 1.

Table 1: Basic angles encountered in trigonometry
θ sinθ cosθ tanθ
0 0 1 0
30 1/2 3/2 1/3
45 2/2 2/2 1
60 3/2 1/2 3
90 1 0

0.2 Other angles by addition and halving

These basic angles can be easily extended to obtain more angles of interest. Adding multiplesMathworldPlanetmathPlanetmath of 90 merely rotates these angles into other quadrantsMathworldPlanetmath; the appropriate values of sin and cos can be obtained through symmetryMathworldPlanetmathPlanetmath.

The values for 15 can be obtained by using the formula for the difference of angles (http://planetmath.org/AngleSumIdentity):

sin15 =sin(45-30)

Likewise, we can find that

cos(15) =6+24
sin(75) =sin(45+30)=6+24
cos(75) =6-24.

More exact angles can be obtained by solving the double angle identity:


So for example, sin7.5=(4-6-2)/8. These angles can be further added and subdivided to obtain a dense subset of exactly known angles. However, such effort is not generally useful. Computers and calculators use a combinationMathworldPlanetmath of lookup-tables and numeric iteration to obtain their values.

0.3 The angles 18, 36, 54, 72

The 18-36-54-72 series of angles cannot be obtained by halving, doubling, adding or subtracting the previous angles. Nevertheless, they are constructible, and their exact values can be derived by the following elementary procedure:

Consider an isosceles triangleMathworldPlanetmath with the angles 72, 54 and 54. From the triangle we derive the relation:


Notice that 72=4×18 and 54=3×18, so if x=18, then

sin2x =cos3x
2sinxcosx =4cos3x-3cosx
2sinx =4cos2x-3
2sinx =4(1-sin2x)-3

The last equation is a quadratic equation that can be solved for sin18. Carrying out the calculations, we obtain the values in Table 2.

Table 2: Other constructible angles in trigonometry
θ sinθ cosθ
18 5-14 5+522
36 5-522 5+14
54 5+14 5-522
72 5+522 5-14
Title exact trigonometry tables
Canonical name ExactTrigonometryTables
Date of creation 2013-03-22 15:31:26
Last modified on 2013-03-22 15:31:26
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 11
Author stevecheng (10074)
Entry type Example
Classification msc 51M04
Classification msc 33B10
Related topic ConstructibleAnglesWithIntegerValuesInDegrees
Related topic Cotangent2
Related topic Trigonometry
Related topic TheoremOnConstructibleAngles
Related topic GraphOfEquationXyConstant