16. Use the identity for \( \sin (x-y) \) to show that sine is an odd function (i.e., \( \sin (-x)=-\sin (x)) \). Hint: let \( x=0^{\circ} \).

To demonstrate that sine is an odd function using the identity for \( \sin (x-y) \), we can start by recognizing the identity itself: \[ \sin(x - y) = \sin x \cos y - \cos x \sin y. \] Now, set \( x = 0^\circ \) and \( y = x \). This gives us \( \sin(0^\circ - x) = \sin(-x) \). By applying the identity: \[ \sin(-x) = \sin(0^\circ) \cos x - \cos(0^\circ) \sin x = 0 \cdot \cos x - 1 \cdot \sin x = -\sin x. \] Thus, we find that \( \sin(-x) = -\sin x \), confirming that sine is indeed an odd function. In ancient Greece, the study of trigonometry began with the Greeks using geometry to explore the relationships among the angles and sides of triangles. The Sine function originated from the study of chords in circles, where the length of a chord was related to the angle that subtended at the center. As this knowledge spread through cultures such as India and the Islamic Golden Age, the sine function was formalized and became crucial in astronomy and navigation. In modern applications, the sine function appears everywhere! From the design of sound waves in music to the modeling of tides, sine waves are used to describe periodic phenomena. Engineers rely on these functions when analyzing signals and oscillations. Additionally, in computer graphics, sine is used for simulating realistic motion, such as the swinging of a pendulum or the rolling of waves, illustrating its importance in both theoretical and applied contexts.