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# Sine

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.

## Contents

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) [1]

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 xcos 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 xcos 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 xcos 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

${\displaystyle {\begin{array}{l}\displaystyle {\sin x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }\end{array}}}$

(...)