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exact trigonometry tables
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(Example)
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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 trigonometry.
The basic angles are given in Table 1.
Table 1: Basic angles encountered in trigonometry
| $\theta$ |
$\sin \theta $ |
$\cos \theta$ |
$\tan \theta $ |
| $0^\circ$ |
0 |
1 |
0 |
| $30^\circ$ |
1/2 |
$\sqrt{3}/2$ |
$1/\sqrt{3}$ |
| $45^\circ$ |
$\sqrt{2}/2$ |
$\sqrt{2}/2$ |
1 |
| $60^\circ$ |
$\sqrt{3}/2$ |
1/2 |
$\sqrt{3}$ |
| $90^\circ$ |
1 |
0 |
$\infty$ |
These basic angles can be easily extended to obtain more angles of interest. Adding multiples of $90^\circ$ merely rotates these angles into other quadrants; the appropriate values of $\sin$ and $\cos$ can be obtained through symmetry.
The values for $15^\circ$ can be obtained by using the formula for the difference of angles:
Likewise, we can find that
More exact angles can be obtained by solving the double angle identity:$$ \sin \frac{\theta}{2} =\pm\sqrt{\frac{1-\cos \theta}{2}}\,, \quad \cos \frac{\theta}{2} =\pm\sqrt{\frac{1+\cos \theta}{2}}\,.$$ So for example, $\sin{7.5^\circ}=\sqrt{(4-\sqrt{6}-\sqrt{2})/8}$ . 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 combination of
lookup-tables and numeric iteration to obtain their values.
The $18^\circ$ -$36^\circ$ -$54^\circ$ -$72^\circ$ 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 triangle with the angles $72^\circ$ , $54^\circ$ and $54^\circ$ . From the triangle we derive the relation:$$ \sin \frac{72^\circ}{2} = \cos 54^\circ$$ Notice that $72=4 \times 18$ and $54 = 3\times 18$ , so if $x = 18^\circ$ , then
The last equation is a quadratic equation that can be solved for $\sin 18^\circ$ . Carrying out the calculations, we obtain the values in Table 2.
Table 2: Other constructible angles in trigonometry
| $\theta$ |
$\sin \theta $ |
$\cos \theta $ |
| $18^\circ$ |
$\dfrac{\sqrt{5}-1}{4}$ |
$\dfrac{\sqrt{5 + \sqrt{5}}}{2\sqrt{2}}$ |
| $36^\circ$ |
$\dfrac{\sqrt{5-\sqrt{5}}}{2\sqrt{2}}$ |
$\dfrac{\sqrt{5}+1}{4}$ |
| $54^\circ$ |
$\dfrac{\sqrt{5}+1}{4}$ |
$\dfrac{\sqrt{5-\sqrt{5}}}{2\sqrt{2}}$ |
| $72^\circ$ |
$\dfrac{\sqrt{5+\sqrt{5}}}{2\sqrt{2}}$ |
$\dfrac{\sqrt{5}-1}{4}$ |
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"exact trigonometry tables" is owned by stevecheng. [ full author list (2) | owner history (1) ]
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Cross-references: quadratic equation, equation, relation, triangle, isosceles triangle, constructible, series, iteration, combination, calculators, computers, dense subset, double angle identity, symmetry, quadrants, rotates, multiples, trigonometry, algebraic, closed, angles
There is 1 reference to this entry.
This is version 8 of exact trigonometry tables, born on 2005-10-02, modified 2007-06-30.
Object id is 7395, canonical name is ExactTrigonometryTables.
Accessed 11787 times total.
Classification:
| AMS MSC: | 33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions) | | | 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries) |
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Pending Errata and Addenda
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