# Sine Graph

A graph between angles and associated trigonometric function sine values is defined sine graph.

The construction of a graph between angles and associated trigonometric ratio sine values, acts as a perfect guide to study the functionality of the sine function. It explains how it responses to various angles and also helps us to understand us its mathematical properties.

## Explanation

Assume $x$ is an angle of a right angle triangle and its corresponding trigonometric function is $\mathrm{sin}x$. Give various values to trigonometric ratio $\mathrm{sin}x$ , and it returns corresponding values. Plot a graph by taking angles (both positive and negative angles) in horizontal axis and associated sine values in vertical axis. It gives a smooth oscillated wave, called sine wave as displayed below.

Trigonometric function sine follows a pattern, and the same pattern can be observed in this graph repeatedly due to the period.

### Analysis

The values of sine function for angles from 0 to 2π (means 360°) is calculated and given here in tabs based tabular form.

Sine values for Postive Angles (from 0° - 90°)
$x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$
0.000000000 23° 0.390731128 46° 0.719339800 69° 0.933580426
0.017452406 24° 0.406736643 47° 0.731353702 70° 0.939692621
0.034899497 25° 0.422618262 48° 0.743144825 71° 0.945518576
0.052335956 26° 0.438371147 49° 0.754709580 72° 0.951056516
0.069756474 27° 0.453990500 50° 0.766044443 73° 0.956304756
0.087155743 28° 0.469471563 51° 0.777145961 74° 0.961261696
0.104528463 29° 0.484809620 52° 0.788010754 75° 0.965925826
0.121869343 30° 0.500000000 53° 0.798635510 76° 0.970295726
0.139173101 31° 0.515038075 54° 0.809016994 77° 0.974370065
0.156434465 32° 0.529919264 55° 0.819152044 78° 0.978147601
10° 0.173648178 33° 0.544639035 56° 0.829037573 79° 0.981627183
11° 0.190808995 34° 0.559192903 57° 0.838670568 80° 0.984807753
12° 0.207911691 35° 0.573576436 58° 0.848048096 81° 0.987688341
13° 0.224951054 36° 0.587785252 59° 0.857167301 82° 0.990268069
14° 0.241921896 37° 0.601815023 60° 0.866025404 83° 0.992546152
15° 0.258819045 38° 0.615661475 61° 0.874619707 84° 0.994521895
16° 0.275637356 39° 0.629320391 62° 0.882947593 85° 0.996194698
17° 0.292371705 40° 0.642787610 63° 0.891006524 86° 0.997564050
18° 0.309016994 41° 0.656059029 64° 0.898794046 87° 0.998629535
19° 0.325568154 42° 0.669130606 65° 0.906307787 88° 0.999390827
20° 0.342020143 43° 0.681998360 66° 0.913545458 89° 0.999847695
21° 0.358367950 44° 0.694658370 67° 0.920504853 90° 1.000000000
22° 0.374606593 45° 0.707106781 68° 0.927183855
Sine values for Postive Angles (from 91° - 180°)
$x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$
91° 0.999847695 114° 0.913545458 137° 0.681998360 160° 0.342020143
92° 0.999390827 115° 0.906307787 138° 0.669130606 161° 0.325568154
93° 0.998629535 116° 0.898794046 139° 0.656059029 162° 0.309016994
94° 0.997564050 117° 0.891006524 140° 0.642787610 163° 0.292371705
95° 0.996194698 118° 0.882947593 141° 0.629320391 164° 0.275637356
96° 0.994521895 119° 0.874619707 142° 0.615661475 165° 0.258819045
97° 0.992546152 120° 0.866025404 143° 0.601815023 166° 0.241921896
98° 0.990268069 121° 0.857167301 144° 0.587785252 167° 0.224951054
99° 0.987688341 122° 0.848048096 145° 0.573576436 168° 0.207911691
100° 0.984807753 123° 0.838670568 146° 0.559192903 169° 0.190808995
101° 0.981627183 124° 0.829037573 147° 0.544639035 170° 0.173648178
102° 0.978147601 125° 0.819152044 148° 0.529919264 171° 0.156434465
103° 0.974370065 126° 0.809016994 149° 0.515038075 172° 0.139173101
104° 0.970295726 127° 0.798635510 150° 0.500000000 173° 0.121869343
105° 0.965925826 128° 0.788010754 151° 0.484809620 174° 0.104528463
106° 0.961261696 129° 0.777145961 152° 0.469471563 175° 0.087155743
107° 0.956304756 130° 0.766044443 153° 0.453990500 176° 0.069756474
108° 0.951056516 131° 0.754709580 154° 0.438371147 177° 0.052335956
109° 0.945518576 132° 0.743144825 155° 0.422618262 178° 0.034899497
110° 0.939692621 133° 0.731353702 156° 0.406736643 179° 0.017452406
111° 0.933580426 134° 0.719339800 157° 0.390731128 180° 0.000000000
112° 0.927183855 135° 0.707106781 158° 0.374606593
113° 0.920504853 136° 0.694658370 159° 0.358367950
Sine values for Postive Angles (from 181° - 270°)
$x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$
181° -0.017452406 204° -0.406736643 227° -0.731353702 250° -0.939692621
182° -0.034899497 205° -0.422618262 228° -0.743144825 251° -0.945518576
183° -0.052335956 206° -0.438371147 229° -0.754709580 252° -0.951056516
184° -0.069756474 207° -0.453990500 230° -0.766044443 253° -0.956304756
185° -0.087155743 208° -0.469471563 231° -0.777145961 254° -0.961261696
186° -0.104528463 209° -0.484809620 232° -0.788010754 255° -0.965925826
187° -0.121869343 210° -0.500000000 233° -0.798635510 256° -0.970295726
188° -0.139173101 211° -0.515038075 234° -0.809016994 257° -0.974370065
189° -0.156434465 212° -0.529919264 235° -0.819152044 258° -0.978147601
190° -0.173648178 213° -0.544639035 236° -0.829037573 259° -0.981627183
191° -0.190808995 214° -0.559192903 237° -0.838670568 260° -0.984807753
192° -0.207911691 215° -0.573576436 238° -0.848048096 261° -0.987688341
193° -0.224951054 216° -0.587785252 239° -0.857167301 262° -0.990268069
194° -0.241921896 217° -0.601815023 240° -0.866025404 263° -0.992546152
195° -0.258819045 218° -0.615661475 241° -0.874619707 264° -0.994521895
196° -0.275637356 219° -0.629320391 242° -0.882947593 265° -0.996194698
197° -0.292371705 220° -0.642787610 243° -0.891006524 266° -0.997564050
198° -0.309016994 221° -0.656059029 244° -0.898794046 267° -0.998629535
199° -0.325568154 222° -0.669130606 245° -0.906307787 268° -0.999390827
200° -0.342020143 223° -0.681998360 246° -0.913545458 269° -0.999847695
201° -0.358367950 224° -0.694658370 247° -0.920504853 270° -1.000000000
202° -0.374606593 225° -0.707106781 248° -0.927183855
203° -0.390731128 226° -0.719339800 249° -0.933580426
Sine values for Postive Angles (from 271° - 360°)
$x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$
271° -0.999847695 294° -0.913545458 317° -0.681998360 340° -0.342020143
272° -0.999390827 295° -0.906307787 318° -0.669130606 341° -0.325568154
273° -0.998629535 296° -0.898794046 319° -0.656059029 342° -0.309016994
274° -0.997564050 297° -0.891006524 320° -0.642787610 343° -0.292371705
275° -0.996194698 298° -0.882947593 321° -0.629320391 344° -0.275637356
276° -0.994521895 299° -0.874619707 322° -0.615661475 345° -0.258819045
277° -0.992546152 300° -0.866025404 323° -0.601815023 346° -0.241921896
278° -0.990268069 301° -0.857167301 324° -0.587785252 347° -0.224951054
279° -0.987688341 302° -0.848048096 325° -0.573576436 348° -0.207911691
280° -0.984807753 303° -0.838670568 326° -0.559192903 349° -0.190808995
281° -0.981627183 304° -0.829037573 327° -0.544639035 350° -0.173648178
282° -0.978147601 305° -0.819152044 328° -0.529919264 351° -0.156434465
283° -0.974370065 306° -0.809016994 329° -0.515038075 352° -0.139173101
284° -0.970295726 307° -0.798635510 330° -0.500000000 353° -0.121869343
285° -0.965925826 308° -0.788010754 331° -0.484809620 354° -0.104528463
286° -0.961261696 309° -0.777145961 332° -0.469471563 355° -0.087155743
287° -0.956304756 310° -0.766044443 333° -0.453990500 356° -0.069756474
288° -0.951056516 311° -0.754709580 334° -0.438371147 357° -0.052335956
289° -0.945518576 312° -0.743144825 335° -0.422618262 358° -0.034899497
290° -0.939692621 313° -0.731353702 336° -0.406736643 359° -0.017452406
291° -0.933580426 314° -0.719339800 337° -0.390731128 360° 0.0000000000
292° -0.927183855 315° -0.707106781 338° -0.374606593
293° -0.920504853 316° -0.694658370 339° -0.358367950

Construct a graph by taking listed angles in horizontal axis and associated sine values in vertical axis.

These values start sine wave at origin of graph, bring it to peak value and once again bring back to horizontal axis level at angle π. After that, sine function returns the negative values, creates the same path in negative direction until 2π. After 2π, the graph follows the path same as the previous because, whatever values sine function has given from 0° to 360°, same values are repeated for every 2π interval.

Sine function is an odd function naturally. So, it returns the same values but with negative sign.

Sine values for Postive Angles (from 0° - (-90°))
$x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$
0.000000000 -23° -0.390731128 -46° -0.719339800 -69° -0.933580426
-1° -0.017452406 -24° -0.406736643 -47° -0.731353702 -70° -0.939692621
-2° -0.034899497 -25° -0.422618262 -48° -0.743144825 -71° -0.945518576
-3° -0.052335956 -26° -0.438371147 -49° -0.754709580 -72° -0.951056516
-4° -0.069756474 -27° -0.453990500 -50° -0.766044443 -73° -0.956304756
-5° -0.087155743 -28° -0.469471563 -51° -0.777145961 -74° -0.961261696
-6° -0.104528463 -29° -0.484809620 -52° -0.788010754 -75° -0.965925826
-7° -0.121869343 -30° -0.500000000 -53° -0.798635510 -76° -0.970295726
-8° -0.139173101 -31° -0.515038075 -54° -0.809016994 -77° -0.974370065
-9° -0.156434465 -32° -0.529919264 -55° -0.819152044 -78° -0.978147601
-10° -0.173648178 -33° -0.544639035 -56° -0.829037573 -79° -0.981627183
-11° -0.190808995 -34° -0.559192903 -57° -0.838670568 -80° -0.984807753
-12° -0.207911691 -35° -0.573576436 -58° -0.848048096 -81° -0.987688341
-13° -0.224951054 -36° -0.587785252 -59° -0.857167301 -82° -0.990268069
-14° -0.241921896 -37° -0.601815023 -60° -0.866025404 -83° -0.992546152
-15° -0.258819045 -38° -0.615661475 -61° -0.874619707 -84° -0.994521895
-16° -0.275637356 -39° -0.629320391 -62° -0.882947593 -85° -0.996194698
-17° -0.292371705 -40° -0.642787610 -63° -0.891006524 -86° -0.997564050
-18° -0.309016994 -41° -0.656059029 -64° -0.898794046 -87° -0.998629535
-19° -0.325568154 -42° -0.669130606 -65° -0.906307787 -88° -0.999390827
-20° -0.342020143 -43° -0.681998360 -66° -0.913545458 -89° -0.999847695
-21° -0.358367950 -44° -0.694658370 -67° -0.920504853 -90° -1.000000000
-22° -0.374606593 -45° -0.707106781 -68° -0.927183855
Sine values for Postive Angles (from (-91°) - (-180°))
$x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$
-91° -0.999847695 -114° -0.913545458 -137° -0.681998360 -160° -0.342020143
-92° -0.999390827 -115° -0.906307787 -138° -0.669130606 -161° -0.325568154
-93° -0.998629535 -116° -0.898794046 -139° -0.656059029 -162° -0.309016994
-94° -0.997564050 -117° -0.891006524 -140° -0.642787610 -163° -0.292371705
-95° -0.996194698 -118° -0.882947593 -141° -0.629320391 -164° -0.275637356
-96° -0.994521895 -119° -0.874619707 -142° -0.615661475 -165° -0.258819045
-97° -0.992546152 -120° -0.866025404 -143° -0.601815023 -166° -0.241921896
-98° -0.990268069 -121° -0.857167301 -144° -0.587785252 -167° -0.224951054
-99° -0.987688341 -122° -0.848048096 -145° -0.573576436 -168° -0.207911691
-100° -0.984807753 -123° -0.838670568 -146° -0.559192903 -169° -0.190808995
-101° -0.981627183 -124° -0.829037573 -147° -0.544639035 -170° -0.173648178
-102° -0.978147601 -125° -0.819152044 -148° -0.529919264 -171° -0.156434465
-103° -0.974370065 -126° -0.809016994 -149° -0.515038075 -172° -0.139173101
-104° -0.970295726 -127° -0.798635510 -150° -0.500000000 -173° -0.121869343
-105° -0.965925826 -128° -0.788010754 -151° -0.484809620 -174° -0.104528463
-106° -0.961261696 -129° -0.777145961 -152° -0.469471563 -175° -0.087155743
-107° -0.956304756 -130° -0.766044443 -153° -0.453990500 -176° -0.069756474
-108° -0.951056516 -131° -0.754709580 -154° -0.438371147 -177° -0.052335956
-109° -0.945518576 -132° -0.743144825 -155° -0.422618262 -178° -0.034899497
-110° -0.939692621 -133° -0.731353702 -156° -0.406736643 -179° -0.017452406
-111° -0.933580426 -134° -0.719339800 -157° -0.390731128 -180° 0.000000000
-112° -0.927183855 -135° -0.707106781 -158° -0.374606593
-113° -0.920504853 -136° -0.694658370 -159° -0.358367950
Sine values for Postive Angles (from (-181°) - (-270°))
$x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$
-181° 0.017452406 -204° 0.406736643 -227° 0.731353702 -250° 0.939692621
-182° 0.034899497 -205° 0.422618262 -228° 0.743144825 -251° 0.945518576
-183° 0.052335956 -206° 0.438371147 -229° 0.754709580 -252° 0.951056516
-184° 0.069756474 -207° 0.453990500 -230° 0.766044443 -253° 0.956304756
-185° 0.087155743 -208° 0.469471563 -231° 0.777145961 -254° 0.961261696
-186° 0.104528463 -209° 0.484809620 -232° 0.788010754 -255° 0.965925826
-187° 0.121869343 -210° 0.500000000 -233° 0.798635510 -256° 0.970295726
-188° 0.139173101 -211° 0.515038075 -234° 0.809016994 -257° 0.974370065
-189° 0.156434465 -212° 0.529919264 -235° 0.819152044 -258° 0.978147601
-190° 0.173648178 -213° 0.544639035 -236° 0.829037573 -259° 0.981627183
-191° 0.190808995 -214° 0.559192903 -237° 0.838670568 -260° 0.984807753
-192° 0.207911691 -215° 0.573576436 -238° 0.848048096 -261° 0.987688341
-193° 0.224951054 -216° 0.587785252 -239° 0.857167301 -262° 0.990268069
-194° 0.241921896 -217° 0.601815023 -240° 0.866025404 -263° 0.992546152
-195° 0.258819045 -218° 0.615661475 -241° 0.874619707 -264° 0.994521895
-196° 0.275637356 -219° 0.629320391 -242° 0.882947593 -265° 0.996194698
-197° 0.292371705 -220° 0.642787610 -243° 0.891006524 -266° 0.997564050
-198° 0.309016994 -221° 0.656059029 -244° 0.898794046 -267° 0.998629535
-199° 0.325568154 -222° 0.669130606 -245° 0.906307787 -268° 0.999390827
-200° 0.342020143 -223° 0.681998360 -246° 0.913545458 -269° 0.999847695
-201° 0.358367950 -224° 0.694658370 -247° 0.920504853 -270° 1.000000000
-202° 0.374606593 -225° 0.707106781 -248° 0.927183855
-203° 0.390731128 -226° 0.719339800 -249° 0.933580426
Sine values for Postive Angles (from (-271°) - (-360°))
$x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$ $x$ $\mathrm{sin}x$
-271° 0.999847695 -294° 0.913545458 -317° 0.681998360 -340° 0.342020143
-272° 0.999390827 -295° 0.906307787 -318° 0.669130606 -341° 0.325568154
-273° 0.998629535 -296° 0.898794046 -319° 0.656059029 -342° 0.309016994
-274° 0.997564050 -297° 0.891006524 -320° 0.642787610 -343° 0.292371705
-275° 0.996194698 -298° 0.882947593 -321° 0.629320391 -344° 0.275637356
-276° 0.994521895 -299° 0.874619707 -322° 0.615661475 -345° 0.258819045
-277° 0.992546152 -300° 0.866025404 -323° 0.601815023 -346° 0.241921896
-278° 0.990268069 -301° 0.857167301 -324° 0.587785252 -347° 0.224951054
-279° 0.987688341 -302° 0.848048096 -325° 0.573576436 -348° 0.207911691
-280° 0.984807753 -303° 0.838670568 -326° 0.559192903 -349° 0.190808995
-281° 0.981627183 -304° 0.829037573 -327° 0.544639035 -350° 0.173648178
-282° 0.978147601 -305° 0.819152044 -328° 0.529919264 -351° 0.156434465
-283° 0.974370065 -306° 0.809016994 -329° 0.515038075 -352° 0.139173101
-284° 0.970295726 -307° 0.798635510 -330° 0.500000000 -353° 0.121869343
-285° 0.965925826 -308° 0.788010754 -331° 0.484809620 -354° 0.104528463
-286° 0.961261696 -309° 0.777145961 -332° 0.469471563 -355° 0.087155743
-287° 0.956304756 -310° 0.766044443 -333° 0.453990500 -356° 0.069756474
-288° 0.951056516 -311° 0.754709580 -334° 0.438371147 -357° 0.052335956
-289° 0.945518576 -312° 0.743144825 -335° 0.422618262 -358° 0.034899497
-290° 0.939692621 -313° 0.731353702 -336° 0.406736643 -359° 0.017452406
-291° 0.933580426 -314° 0.719339800 -337° 0.390731128 -360° 0.0000000000
-292° 0.927183855 -315° 0.707106781 -338° 0.374606593
-293° 0.920504853 -316° 0.694658370 -339° 0.358367950

Compare values of sine function for negative angles with values for positive angles. Both are absolutely same but signs are changed. Thence, the graph displays same path but in reverse direction due to negative sign of values.

Combine both graphs and extend the graph by taking numerous angles and associated sine values to see the oscillated sine wave.

For every 2π interval, the path will be repeated, so the interval 2π is called period of the sine wave. Sine function is able to give value to any angle. According to values of sine function and also from graph, it is understandable that it can only give maximum 1 and minimum -1. It oscillates between -1 and 1 possibly.