Exact trigonometric values
Introduction
Exact and approximate values of some trigonometric functions.
Notes:
- The formulas are sorted first by function, then by the approximate “complexity” (usually length) of the formula, then by angle.
- In the formulas, √a+b means √(a)+b, not √(a+b).
- The approximate values have been rounded to nine digits after the decimal point.
- The formulas should be readable in a text browser.
Sines and cosines
The table contains these angles between 0°…90°:
- multiples of 3°
- multiples of 3.75° (15°/4)
- multiples of 4.5° (9°/2)
- multiples of 5.625° (45°/8)
Notes:
- sin(α) = cos(90°−α)
- cos(α) = sin(90°−α)
- cos(α/2) = √((cos(α)+1)/2)
- cos(2α) = 2×cos(α)2
- cos(3α) = cos(α) × (4×cos(α)2−3)
| Sine of | Cosine of | Formula | Approximate value |
|---|---|---|---|
| 0° | 90° | 0 | 0.000000000 |
| 90° | 0° | 1 | 1.000000000 |
| 30° | 60° | 1 / 2 | 0.500000000 |
| 45° | 45° | √2 / 2 | 0.707106781 |
| 60° | 30° | √3 / 2 | 0.866025404 |
| 18° | 72° | (−1 + √5) / 4 | 0.309016994 |
| 54° | 36° | ( 1 + √5) / 4 | 0.809016994 |
| 15° | 75° | (√6 − √2) / 4 | 0.258819045 |
| 75° | 15° | (√6 + √2) / 4 | 0.965925826 |
| 22.5° | 67.5° | √(2 − √2) / 2 | 0.382683432 |
| 67.5° | 22.5° | √(2 + √2) / 2 | 0.923879533 |
| 36° | 54° | √(10 − 2√5) / 4 | 0.587785252 |
| 72° | 18° | √(10 + 2√5) / 4 | 0.951056516 |
| 11.25° | 78.75° | √(2 − √(2 + √2)) / 2 | 0.195090322 |
| 33.75° | 56.25° | √(2 − √(2 − √2)) / 2 | 0.555570233 |
| 56.25° | 33.75° | √(2 + √(2 − √2)) / 2 | 0.831469612 |
| 78.75° | 11.25° | √(2 + √(2 + √2)) / 2 | 0.980785280 |
| 7.5° | 82.5° | √(2 − √(2 + √3)) / 2 | 0.130526192 |
| 37.5° | 52.5° | √(2 − √(2 − √3)) / 2 | 0.608761429 |
| 52.5° | 37.5° | √(2 + √(2 − √3)) / 2 | 0.793353340 |
| 82.5° | 7.5° | √(2 + √(2 + √3)) / 2 | 0.991444861 |
| 9° | 81° | √(8 − √(40 + 8√5)) / 4 | 0.156434465 |
| 27° | 63° | √(8 − √(40 − 8√5)) / 4 | 0.453990500 |
| 63° | 27° | √(8 + √(40 − 8√5)) / 4 | 0.891006524 |
| 81° | 9° | √(8 + √(40 + 8√5)) / 4 | 0.987688341 |
| 5.625° | 84.375° | √(2 − √(2 + √(2 + √2))) / 2 | 0.098017140 |
| 16.875° | 73.125° | √(2 − √(2 + √(2 − √2))) / 2 | 0.290284677 |
| 28.125° | 61.875° | √(2 − √(2 − √(2 − √2))) / 2 | 0.471396737 |
| 39.375° | 50.625° | √(2 − √(2 − √(2 + √2))) / 2 | 0.634393284 |
| 50.625° | 39.375° | √(2 + √(2 − √(2 + √2))) / 2 | 0.773010453 |
| 61.875° | 28.125° | √(2 + √(2 − √(2 − √2))) / 2 | 0.881921264 |
| 73.125° | 16.875° | √(2 + √(2 + √(2 − √2))) / 2 | 0.956940336 |
| 84.375° | 5.625° | √(2 + √(2 + √(2 + √2))) / 2 | 0.995184727 |
| 3.75° | 86.25° | √(2 − √(2 + √(2 + √3))) / 2 | 0.065403129 |
| 18.75° | 71.25° | √(2 − √(2 + √(2 − √3))) / 2 | 0.321439465 |
| 26.25° | 63.75° | √(2 − √(2 − √(2 − √3))) / 2 | 0.442288690 |
| 41.25° | 48.75° | √(2 − √(2 − √(2 + √3))) / 2 | 0.659345815 |
| 48.75° | 41.25° | √(2 + √(2 − √(2 + √3))) / 2 | 0.751839807 |
| 63.75° | 26.25° | √(2 + √(2 − √(2 − √3))) / 2 | 0.896872742 |
| 71.25° | 18.75° | √(2 + √(2 + √(2 − √3))) / 2 | 0.946930129 |
| 86.25° | 3.75° | √(2 + √(2 + √(2 + √3))) / 2 | 0.997858923 |
| 6° | 84° | (−1 − √5 + √(30 − 6√5)) / 8 | 0.104528463 |
| 42° | 48° | ( 1 − √5 + √(30 + 6√5)) / 8 | 0.669130606 |
| 66° | 24° | ( 1 + √5 + √(30 − 6√5)) / 8 | 0.913545458 |
| 78° | 12° | (−1 + √5 + √(30 + 6√5)) / 8 | 0.978147601 |
| 12° | 78° | √(7 − √5 − √(30 − 6√5)) / 4 | 0.207911691 |
| 24° | 66° | √(7 + √5 − √(30 + 6√5)) / 4 | 0.406736643 |
| 48° | 42° | √(7 − √5 + √(30 − 6√5)) / 4 | 0.743144825 |
| 84° | 6° | √(7 + √5 + √(30 + 6√5)) / 4 | 0.994521895 |
| 4.5° | 85.5° | √(8 − √(32 + √(640 + 128√5))) / 4 | 0.078459096 |
| 13.5° | 76.5° | √(8 − √(32 + √(640 − 128√5))) / 4 | 0.233445364 |
| 31.5° | 58.5° | √(8 − √(32 − √(640 − 128√5))) / 4 | 0.522498565 |
| 40.5° | 49.5° | √(8 − √(32 − √(640 + 128√5))) / 4 | 0.649448048 |
| 49.5° | 40.5° | √(8 + √(32 − √(640 + 128√5))) / 4 | 0.760405966 |
| 58.5° | 31.5° | √(8 + √(32 − √(640 − 128√5))) / 4 | 0.852640164 |
| 76.5° | 13.5° | √(8 + √(32 + √(640 − 128√5))) / 4 | 0.972369920 |
| 85.5° | 4.5° | √(8 + √(32 + √(640 + 128√5))) / 4 | 0.996917334 |
| 3° | 87° | √(8 − √3 − √15 − √(10 − 2√5)) / 4 | 0.052335956 |
| 21° | 69° | √(8 + √3 − √15 − √(10 + 2√5)) / 4 | 0.358367950 |
| 33° | 57° | √(8 − √3 − √15 + √(10 − 2√5)) / 4 | 0.544639035 |
| 39° | 51° | √(8 − √3 + √15 − √(10 + 2√5)) / 4 | 0.629320391 |
| 51° | 39° | √(8 + √3 − √15 + √(10 + 2√5)) / 4 | 0.777145961 |
| 57° | 33° | √(8 + √3 + √15 − √(10 − 2√5)) / 4 | 0.838670568 |
| 69° | 21° | √(8 − √3 + √15 + √(10 + 2√5)) / 4 | 0.933580426 |
| 87° | 3° | √(8 + √3 + √15 + √(10 − 2√5)) / 4 | 0.998629535 |
Tangents
The table contains these angles between 0°…90°:
- multiples of 3°
- multiples of 7.5° (15°/2)
- multiples of 11.25° (45°/4)
Notes:
- tan(α) = cot(90°−α) = sin(α)/cos(α)
- cot(α) = tan(90°−α)
- The formulas for tangents of 3°, 21°, 33°, 39°, 51°, 57°, 69° and 87° were created by me based on the formulas in the “Sines and cosines” section above. I'm not sure if my formulas could be simplified further.
| Tangent of | Formula | Approximate value |
|---|---|---|
| 0° | 0 | 0.000000000 |
| 45° | 1 | 1.000000000 |
| 60° | √3 | 1.732050808 |
| 30° | √3 / 3 | 0.577350269 |
| 22.5° | −1 + √2 | 0.414213562 |
| 67.5° | 1 + √2 | 2.414213562 |
| 15° | 2 − √3 | 0.267949192 |
| 75° | 2 + √3 | 3.732050808 |
| 36° | √(5 − 2√5) | 0.726542528 |
| 72° | √(5 + 2√5) | 3.077683537 |
| 18° | √(25 − 10√5) / 5 | 0.324919696 |
| 54° | √(25 + 10√5) / 5 | 1.376381920 |
| 7.5° | −2 + √6 − √3 + √2 | 0.131652498 |
| 37.5° | −2 + √6 + √3 − √2 | 0.767326988 |
| 52.5° | 2 + √6 − √3 − √2 | 1.303225373 |
| 82.5° | 2 + √6 + √3 + √2 | 7.595754113 |
| 11.25° | −1 − √2 + √(4 + 2√2) | 0.198912367 |
| 33.75° | 1 − √2 + √(4 − 2√2) | 0.668178638 |
| 56.25° | −1 + √2 + √(4 − 2√2) | 1.496605763 |
| 78.75° | 1 + √2 + √(4 + 2√2) | 5.027339492 |
| 9° | 1 + √5 − √(5 + 2√5) | 0.158384440 |
| 27° | −1 + √5 − √(5 − 2√5) | 0.509525449 |
| 63° | −1 + √5 + √(5 − 2√5) | 1.962610506 |
| 81° | 1 + √5 + √(5 + 2√5) | 6.313751515 |
| 6° | √(7 − 2√5 − √(60 − 24√5)) | 0.105104235 |
| 42° | √(7 + 2√5 − √(60 + 24√5)) | 0.900404044 |
| 66° | √(7 − 2√5 + √(60 − 24√5)) | 2.246036774 |
| 78° | √(7 + 2√5 + √(60 + 24√5)) | 4.704630109 |
| 12° | √(23 − 10√5 − √(1020 − 456√5)) | 0.212556562 |
| 24° | √(23 + 10√5 − √(1020 + 456√5)) | 0.445228685 |
| 48° | √(23 − 10√5 + √(1020 − 456√5)) | 1.110612515 |
| 84° | √(23 + 10√5 + √(1020 + 456√5)) | 9.514364454 |
| 3° | √((8 − √3 − √15 − √(10 − 2√5)) / (8 + √3 + √15 + √(10 − 2√5))) | 0.052407779 |
| 21° | √((8 + √3 − √15 − √(10 + 2√5)) / (8 − √3 + √15 + √(10 + 2√5))) | 0.383864035 |
| 33° | √((8 − √3 − √15 + √(10 − 2√5)) / (8 + √3 + √15 − √(10 − 2√5))) | 0.649407593 |
| 39° | √((8 − √3 + √15 − √(10 + 2√5)) / (8 + √3 − √15 + √(10 + 2√5))) | 0.809784033 |
| 51° | √((8 + √3 − √15 + √(10 + 2√5)) / (8 − √3 + √15 − √(10 + 2√5))) | 1.234897157 |
| 57° | √((8 + √3 + √15 − √(10 − 2√5)) / (8 − √3 − √15 + √(10 − 2√5))) | 1.539864964 |
| 69° | √((8 − √3 + √15 + √(10 + 2√5)) / (8 + √3 − √15 − √(10 + 2√5))) | 2.605089065 |
| 87° | √((8 + √3 + √15 + √(10 − 2√5)) / (8 − √3 − √15 − √(10 − 2√5))) | 19.081136688 |
Sources
- Wikipedia: Exact trigonometric values
- Wikimedia Commons: Exact trigonometric table for multiples of 3°
- Exact Values of the Sine and Cosine Functions in Increments of 3 degrees (archived)
- Exact Trigonometric Function Values
- Wolfram Alpha – looks like it won't show the exact values for some formulas unless you write the angle as a fraction:
e.g.
tan(Divide[6°,1])works buttan(6°)won't
