Trigonometry is a fundamental branch of mathematics that explores the relationships between the sides and angles of triangles. It focuses primarily on right triangles, where one angle measures exactly 90 degrees. The word trigonometry comes from Greek, meaning triangle measurement.
The three basic trigonometric functions are sine, cosine, and tangent. Sine is the ratio of the opposite side to the hypotenuse. Cosine is the ratio of the adjacent side to the hypotenuse. Tangent is the ratio of the opposite side to the adjacent side. These ratios remain constant for any given angle, regardless of the triangle's size.
The unit circle provides a powerful way to understand trigonometry. In a unit circle with radius one, any point on the circle has coordinates that directly represent trigonometric values. The x-coordinate equals cosine theta, and the y-coordinate equals sine theta. As the angle changes, we can see how these values change smoothly.
Trigonometric functions create beautiful wave patterns when graphed. 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 pattern but starts at one. Both functions repeat this pattern every two pi radians, making them periodic functions.
Trigonometry has countless real-world applications. Engineers use it to calculate heights and distances in construction projects. Physicists apply it to analyze wave motion and oscillations. Computer graphics rely on trigonometry for rotations and transformations. Navigation systems use it for positioning, and even music theory incorporates trigonometric concepts to understand sound waves and harmonics.