Trigonometry is a fundamental branch of mathematics that studies the relationships between angles and sides in triangles. In a right triangle, we have angle theta, the opposite side b, the adjacent side a, and the hypotenuse c. These relationships form the foundation for understanding trigonometric functions.
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 calculate unknown sides and angles in right triangles.
Trigonometric functions can also be understood using the unit circle. On the unit circle, angle theta corresponds to a point where the x-coordinate equals cosine and the y-coordinate equals sine. As the point moves around the circle, sine and cosine values change continuously. Tangent equals y divided by x.
The graphs of trigonometric functions show their periodic nature. The sine function creates a wave-like pattern, starting at zero, reaching one at pi over two, returning to zero at pi, dropping to negative one at three pi over two, and back to zero at two pi. The cosine function follows a similar pattern but shifted horizontally. Both functions repeat this cycle with a period of two pi.