Trigonometric functions are fundamental mathematical functions that describe the relationships between angles and side lengths in triangles. In a right triangle, we have angle theta, the opposite side b, the adjacent side a, and the hypotenuse c. These functions are essential tools in mathematics, physics, and engineering.
The three basic trigonometric functions are sine, cosine, and tangent. Sine is defined as the opposite side divided by the hypotenuse. Cosine is the adjacent side divided by the hypotenuse. Tangent is the opposite side divided by the adjacent side. These ratios remain constant for any given angle, regardless of the triangle's size.
Trigonometric functions can also be understood through the unit circle. In the unit circle, angle theta corresponds to a point on the circle where the x-coordinate equals cosine and the y-coordinate equals sine. As the point moves around the circle, sine and cosine values change continuously. Tangent equals y divided by x.
The graphs of trigonometric functions reveal their periodic nature. The sine function creates a wave-like pattern, starting at zero, reaching one at pi over two, returning to zero at pi, dropping to negative one at three pi over two, and completing the cycle at two pi. The cosine function follows a similar pattern but is shifted horizontally. Both functions repeat this pattern every two pi units, making them periodic functions.
Trigonometric functions have countless applications across science and engineering. In physics, they describe waves and oscillations. Engineers use them for signal processing and structural analysis. Computer graphics rely on them for rotations and transformations. They are essential in navigation, astronomy, architecture, and construction. These functions provide the mathematical foundation for understanding and modeling any periodic or cyclical phenomenon in nature and technology.