Welcome to trigonometry! Trigonometry is the mathematical study of relationships between angles and sides in triangles. This powerful branch of mathematics helps us solve real-world problems involving heights, distances, periodic phenomena, and wave patterns. Let's start by examining a right triangle with its three sides: the hypotenuse, adjacent side, and opposite side relative to angle theta.
Now let's explore the three fundamental trigonometric functions. Sine is 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 are constant for any given angle, making them incredibly useful for calculations involving triangles of any size.
The unit circle revolutionizes our understanding of trigonometry. On a circle with radius one, any point's coordinates directly give us the cosine and sine values. The x-coordinate equals cosine theta, and the y-coordinate equals sine theta. This powerful concept extends trigonometry beyond right triangles, allowing us to find trigonometric values for any angle, including those greater than ninety degrees.
Trigonometric functions create stunning wave patterns that are fundamental to understanding periodic phenomena. The sine wave starts at zero and oscillates smoothly between negative one and positive one with a period of two pi. The cosine wave has the same shape but starts at one, essentially a sine wave shifted by pi over two. These beautiful patterns appear everywhere in nature, from sound waves to ocean tides, and form the foundation of signal processing in modern technology.