Sine

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The sine function is an elementary transcendental function. The sine of an angle

θ

, denoted as

sin θ

, is one of the most important [circular] trigonometric functions.

Given the angle

θ

of an arc on a unit circle,

sin θ

is the length of the side on a right triangle opposing a vertex coinciding with the center of the circle (the other two sides of the triangle being the hypotenuse and a side that is a line along

x = 0

).

(PLACEHOLDER FOR IMAGE) 

Per the Pythagorean theorem,

(sin θ  ) 2 + (cos θ  ) 2 = 1

, usually written

sin 2 θ + cos 2 θ = 1

(where

sin 2 θ   :=   (sin θ  ) 2

, i.e. not the sine of the sine of

θ

).

The graph of the sine function has given rise to the term “sine wave” to distinguish between sinuous waves that look like this

(sine is in red, cosine in blue) and sawtooth and triangular waves.

Table of sine and cosine values

For the decimal expansions of the sine from 1 to 89 degrees, see A019810 through A019898 (with the A-number being given by 19809 plus the desired number of degrees between 1 and 89—except for 30, 45 and 60 degrees). In the following table,

y = 90  −  x

, and all non-integral values are given to 8 decimal places (click the link for the sequence entry for far greater precision).

x y sin x cos y A-number 1 89 0.01745241 A019810 2 88 0.03489950 A019811 3 87 0.05233596 A019812 4 86 0.06975647 A019813 5 85 0.08715574 A019814 6 84 0.10452846 A019815 7 83 0.12186934 A019816 8 82 0.13917310 A019817 9 81 0.15643447 A019818 10 80 0.17364818 A019819 11 79 0.19080900 A019820 12 78 0.20791169 A019821 13 77 0.22495105 A019822 14 76 0.24192190 A019823 15 75 0.25881905 A019824 16 74 0.27563736 A019825 17 73 0.29237170 A019826 18 72 0.30901699 A019827 19 71 0.32556815 A019828 20 70 0.34202014 A019829 21 69 0.35836795 A019830 22 68 0.37460659 A019831 23 67 0.39073113 A019832 24 66 0.40673664 A019833 25 65 0.42261826 A019834 26 64 0.43837115 A019835 27 63 0.45399050 A019836 28 62 0.46947156 A019837 29 61 0.48480962 A019838 30 60 0.50000000 A020761

x y sin x cos y A-number 31 59 0.51503807 A019840 32 58 0.52991926 A019841 33 57 0.54463904 A019842 34 56 0.55919290 A019843 35 55 0.57357644 A019844 36 54 0.58778525 A019845 37 53 0.60181502 A019846 38 52 0.61566148 A019847 39 51 0.62932039 A019848 40 50 0.64278761 A019849 41 49 0.65605903 A019850 42 48 0.66913061 A019851 43 47 0.68199836 A019852 44 46 0.69465837 A019853 45 45 0.70710678 A010503 46 44 0.71933980 A019855 47 43 0.73135370 A019856 48 42 0.74314483 A019857 49 41 0.75470958 A019858 50 40 0.76604444 A019859 51 39 0.77714596 A019860 52 38 0.78801075 A019861 53 37 0.79863551 A019862 54 36 0.80901699 A019863 55 35 0.81915204 A019864 56 34 0.82903757 A019865 57 33 0.83867057 A019866 58 32 0.84804810 A019867 59 31 0.85716730 A019868 60 30 0.86602540 A010527

x y sin x cos y A-number 61 29 0.87461971 A019870 62 28 0.88294759 A019871 63 27 0.89100652 A019872 64 26 0.89879405 A019873 65 25 0.90630779 A019874 66 24 0.91354546 A019875 67 23 0.92050485 A019876 68 22 0.92718385 A019877 69 21 0.93358043 A019878 70 20 0.93969262 A019879 71 19 0.94551858 A019880 72 18 0.95105652 A019881 73 17 0.95630476 A019882 74 16 0.96126170 A019883 75 15 0.96592583 A019884 76 14 0.97029573 A019885 77 13 0.97437006 A019886 78 12 0.97814760 A019887 79 11 0.98162718 A019888 80 10 0.98480775 A019889 81 9 0.98768834 A019890 82 8 0.99026807 A019891 83 7 0.99254615 A019892 84 6 0.99452190 A019893 85 5 0.99619470 A019894 86 4 0.99756405 A019895 87 3 0.99862953 A019896 88 2 0.99939083 A019897 89 1 0.99984770 A019898 90 0 1.00000000 A000007

Taylor series expansion

The Taylor series expansion of the sine function is

sin x  =    ∞ ∑   n  = 0    (−1) n (2 n + 1)! x 2 n + 1  =  x  −      x 3 3!   +     x 5 5!   −     x 7 7!   +   ⋯.

Formulae

(...)

See also

• {{sin}} mathematical function template