Draw a circle of radius 1 , and pick a point on the circumference . Then draw a straight line through this point and the origin . Call the angle it makes with the origin a . Then the y coordinate of your point is sina and the x coordinate is cosa . Another way of saying this is that sin and cos parameterize the unit circle . Alternatively you can think of them as the ratios of side lengths in a right triangle this is where the term trigonometry comes from . That ' s the geometric interpretation . If you want a more rigorous definition , they can be defined in terms of power series as uaccidentallybrill said you do need to show that these series converge , but they do . You can also write them in terms of complex exponentialssinxeix eix 2 cosxeix eix 2