Trigonometry is a fundamental branch of mathematics that studies the relationships between angles and sides in triangles. It focuses primarily on right-angled triangles, where we can define three main sides: the hypotenuse, which is the longest side opposite the right angle, the adjacent side next to our angle of interest, and the opposite side across from our angle.
The three fundamental trigonometric functions are sine, cosine, and tangent. Sine is defined as the ratio of the opposite side to the hypotenuse. Cosine is the ratio of the adjacent side to the hypotenuse. And 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 trigonometric functions. In a unit circle with radius one, any point on the circle has coordinates that directly represent cosine and sine 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.
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 and is shifted by pi over two. Both functions repeat this pattern every two pi units, making them periodic functions.
Trigonometry has countless real-world applications. It's used in navigation and GPS systems to determine positions and distances. Engineers and architects use it to calculate forces and design structures. In physics, it helps analyze waves and oscillations. Computer graphics rely on trigonometry for rotations and transformations. It's also essential in signal processing, astronomy, and many other fields, making it one of the most practical branches of mathematics.