Free Sine Curve Essay Sample

The sine is the ratio of the opposite leg of the right triangle to its hypotenuse. Let’s imagine that we have the Cartesian coordinate system and a circumference with the same center and radius R. Every angle can be depicted as a rotation operation from the positive x-axis to some interval that joins together the center of the Cartesian coordinate system and some point of the circumference. The length of this interval is equal to the radius and can be depicted as the hypotenuse of the right triangle. The adjacent leg of the triangle belongs to the x-axis, and its opposite leg is a vertical that joins the point of the circumference with some point of the x-axis. It can be clearly seen on the interactive unit circle: the adjacent leg is marked with blue, and the opposite leg is red.
The sine of 0° and 180° is 0, as the length of the opposite leg of the triangle in this points is 0. The sine of 90° is 1, as the length of the opposite leg of the triangle is equal to the length of the hypotenuse and R. The sine of 270° is -1, as the function of sine is uneven, and sin (-x) = - sin (x). Thus, sin (270°) = sin (-90°) = - sin (90°) = -1. The sine of 360° is equal to the sine of 0° and is 0. We get the four main points of the sine curve. These valuations and their origination can be seen on the interactive unit circle and on the sine curve. The sine curve is located within the limits of [-R; R] in the Y-direction. The calculation of valuations of the sine in the feature points within the limits of [0; π/2] (30°, 45°, and 60°) allows constructing the quarter of the curve. The quarter of the curve within the limits of [π/2; π] is symmetric, as sin (x) = sin (π/2 + ϕ) = sin (π/2 - ϕ), 0≤ϕ≤ π/2. The second half of the sine curve is constructed due to its unevenness.
The interactive unit circle graphically shows why the sine function has specific valuations in its main points and helps to understand how the sine curve can be constructed. The quarter of the sine curve within the limits of [π/2; π] is symmetric to the one within the limits of [0; π/2], and the half of the curve within the limits of [π; 2π] is inversely symmetric to its first half due to the unevenness of the sine function.