Trigfunctions

Trigonometric functions, often shortened to trig functions, are fundamental mathematical functions that relate an angle of a right-angled triangle to the ratios of the lengths of its sides. The most common trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions are defined based on the angles and sides of a right-angled triangle. For an acute angle in a right-angled triangle, the sine is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine is the ratio of the length of the adjacent side to the hypotenuse. The tangent is the ratio of the length of the opposite side to the adjacent side.

Beyond right-angled triangles, trigonometric functions are extended to all real numbers using the unit circle. In this context, the angle is measured from the positive x-axis, and the sine and cosine of the angle correspond to the y and x coordinates, respectively, of the point where the terminal side of the angle intersects the unit circle. The tangent is then the ratio of the sine to the cosine.

Trigonometric functions have a periodic nature, meaning their values repeat at regular intervals. This periodicity is crucial in describing phenomena that exhibit cyclical behavior, such as waves, oscillations, and rotations. They are essential tools in various fields, including physics, engineering, astronomy, navigation, and signal processing, where they are used to model and analyze wave phenomena, periodic motion, and geometric relationships.