Sine And Cosine Essays Example

1. The Sine and Cosine graphs are 'complementary', which means that the expression sin2 x + cos2 x = 1 is realized. Therefore, when sin (x) = 1, in that point cos(x) = 0, and vice versa. The same situation is observed when the functions take -1 and 0 values. Therefore, if Sine and Cosine are plotted on the same graphs, they look the same, but as if Cosine function is shifted by pi/2 comparing to Sine.
Since Tangent function is obtained as sin x / cos x, then the point x = it should not reach the x-points pi/2, 3pi/2, etc. The Tangent is negative when sin x = 0, which is 0, pi, 2pi, etc. Therefore, it looks as a series of the curves which slide down from infinite to zero, and then continues slide to the negative side and asymptotically approaches imaginary lines through pi/2, 3pi/2, etc.
2. The graphs discussed also presents another interesting feature of the function. This is the periodic behavior of the Sine function. The same way, basing on the understanding gained in (1), Cosine is also a periodical function. As the function moves further the x-axis, it forms a triangle, which provides a value of Sine function by division of the opposite cathetus to hypotenuse. By the moving around the cirle, it is obvious that sin x function is finite, e.g. it may exist in all the possible x-values. To generate a function, the point moves around the circle and builds the relation for this function. One run of the circle equals pi, which is a period of sin x, and in all the consequential evenings the value sin x takes values from 0 to 1. Here, we can see that the function is periodical and the values repeat after the period pi has been reached. It is a useful tool for understanding real-life environmental protection.