Trigonometric functions are fundamental tools in mathematics that help us understand the relationship between angles and side lengths in right triangles. The three main functions are sine, cosine, and tangent, each representing a specific ratio of triangle sides.
The three basic trigonometric functions can be remembered using the mnemonic SOH CAH TOA. Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent. These ratios remain constant for any given angle, regardless of the triangle's size.
Trigonometric functions extend beyond right triangles using the unit circle. On a circle with radius one, any point corresponds to an angle theta. The x-coordinate of this point equals cosine theta, and the y-coordinate equals sine theta. This allows us to define trigonometric functions for any angle, not just those between zero and ninety degrees.
The graphs of trigonometric functions reveal their periodic nature. The sine function starts at zero, rises to one at pi over two, returns to zero at pi, drops to negative one at three pi over two, and completes the cycle at two pi. The cosine function follows a similar wave pattern but starts at one. Both functions repeat this pattern every two pi units, making them periodic functions.
Trigonometric functions have countless practical applications. Engineers use them to analyze forces and design structures. In physics, they describe wave motion and oscillations. Navigation systems rely on trigonometry for GPS calculations. Computer graphics use these functions to create realistic animations and 3D models. The fundamental identity sine squared plus cosine squared equals one is essential in many mathematical proofs and calculations.