Trigonometry is the branch of mathematics that studies relationships between angles and sides of triangles. It focuses on right triangles and uses special functions to connect angles with side ratios. In a right triangle, we have an angle theta, the opposite side b, the adjacent side a, and the hypotenuse c.
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 help us find unknown sides or angles in right triangles.
The unit circle extends trigonometry beyond right triangles. It's a circle with radius one centered at the origin. For any angle theta on the unit circle, the x-coordinate gives us cosine theta, and the y-coordinate gives us sine theta. This allows us to define trigonometric functions for any angle, not just those in right triangles.
Trigonometric functions create periodic 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 every two pi radians and oscillate between negative one and one.
Trigonometry has countless real-world applications. Navigation and GPS systems use trigonometry to determine positions and distances. Engineers and architects rely on angle calculations for structural design. Physics uses trigonometry to describe waves, oscillations, and periodic motion. Computer graphics and animation depend on trigonometric functions for rotations and transformations. Even music and sound analysis utilize trigonometric principles to understand frequencies and harmonics.